70:
1574:
1504:
1492:
1820:
27:
1287:, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. A great example of perpendicularity can be seen in any compass, note the cardinal points; North, East, South, West (NESW) The line N-S is perpendicular to the line W-E and the angles N-E, E-S, S-W and W-N are all 90Ā° to one another.
1564:
can be used as the basis of methods of constructing right angles. For example, by counting links, three pieces of chain can be made with lengths in the ratio 3:4:5. These can be laid out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out
2893:
of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular. This implies that, seen from any point on its directrix, any parabola subtends a right angle.
2835:
states that two lines both through the same point on a circle but going through opposite endpoints of a diameter are perpendicular. This is equivalent to saying that any diameter of a circle subtends a right angle at any point on the circle, except the two endpoints of the diameter.
1262:; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its
2461:
2314:
2585:
1271:
A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the
1734:
is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point.
1438:
is frequently used in connection with perpendiculars. This usage is exemplified in the top diagram, above, and its caption. The diagram can be in any orientation. The foot is not necessarily at the bottom.
2813:
The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8
1410:
2739:
2635:
2028:
1963:
1360:
1328:
3107:, if squares are constructed externally on the sides of a quadrilateral, the line segments connecting the centers of opposite squares are perpendicular and equal in length.
2124:
2074:
1509:
Construction of the perpendicular to the half-line h from the point P (applicable not only at the end point A, M is freely selectable), animation at the end with pause 10 s
2953:
2713:
2913:
The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P.
2320:
2173:
2165:
30:
The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. The segment AB can be called
2689:
2662:
1623:
In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because
2875:, the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola.
2467:
1419:
if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.
1362:
if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols,
1782:
1535:
Step 2 (green): construct circles centered at A' and B' having equal radius. Let Q and P be the points of intersection of these two circles.
1565:
gardens and fields, where the dimensions are large, and great accuracy is not needed. The chains can be used repeatedly whenever required.
1253:, ā. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes.
1198:
3100:, a line through the midpoint of one side and through the intersection point of the diagonals is perpendicular to the opposite side.
1620:. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.
1627:
are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines
1517:
298:
2852:
are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect the ellipse.
1762:
fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line. Other
1365:
2879:
1751:
1547:
1543:
264:
3145:
1743:
1731:
1705:
2718:
1191:
1145:
751:
210:
20:
2591:
1786:
1968:
1903:
3214:
3128:
3073:
3253:
2925:
1166:
776:
1333:
1301:
2856:
2090:
so that the origin is situated where the lines cross. Then define two displacements along each line,
1184:
2093:
2033:
3093:
2695:
Both proofs are valid for horizontal and vertical lines to the extent that we can let one slope be
2456:{\displaystyle {\vec {r}}_{2}=x_{2}{\hat {x}}+y_{2}{\hat {y}}=x_{2}{\hat {x}}+m_{2}x_{2}{\hat {y}}}
2309:{\displaystyle {\vec {r}}_{1}=x_{1}{\hat {x}}+y_{1}{\hat {y}}=x_{1}{\hat {x}}+m_{1}x_{1}{\hat {y}}}
1778:
1759:
153:
2931:
2698:
3232:
3116:
3005:
1763:
579:
259:
116:
3302:
3104:
2987:
1690:
1678:
655:
366:
244:
129:
2129:
3140:
3015:
1890:
1277:
427:
388:
347:
342:
195:
3236:
3097:
2917:
2667:
2640:
1249:
1095:
1018:
866:
771:
293:
188:
102:
1755:, for the perpendicular distance between two non-parallel lines in three-dimensional space
8:
2975:
2087:
1889:
In the two-dimensional plane, right angles can be formed by two intersected lines if the
1767:
1613:
1561:
1554:
1100:
1044:
957:
811:
791:
716:
606:
477:
467:
330:
205:
200:
183:
158:
146:
98:
93:
74:
2580:{\displaystyle {\vec {r}}_{1}\cdot {\vec {r}}_{2}=\left(1+m_{1}m_{2}\right)x_{1}x_{2}=0}
1503:
1256:
Perpendicularity is one particular instance of the more general mathematical concept of
3239:
College
Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle
2998:
2832:
1617:
1609:
1059:
786:
626:
254:
178:
168:
139:
124:
3237:
2890:
1747:, for the perpendicular distance from the origin to a plane in three-dimensional space
3257:
3019:
1624:
1264:
1130:
918:
896:
821:
680:
406:
335:
227:
173:
134:
1120:
1049:
846:
756:
3287:
How to draw a perpendicular at the endpoint of a ray with compass and straight edge
2883:
2845:
2776:
If the intersection of any two perpendicular chords divides one chord into lengths
2770:
2083:
1797:
1720:
1709:
1586:
1416:
1236:
1216:
1110:
851:
561:
439:
374:
232:
217:
82:
3286:
3280:
1766:
methods using perpendicular distance to measure the quality of a fit exist, as in
1608:), all of the angles formed along the third line are right angles. Therefore, in
3089:
3047:
2983:
1898:
1801:
1698:
1546:
for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the
533:
396:
239:
222:
163:
69:
3274:
1105:
1074:
1008:
856:
801:
736:
2968:
1805:
1793:
1642:
One of the orange-shaded angles is congruent to one of the green-shaded angles.
1573:
1423:
1295:
1273:
1161:
1069:
1013:
978:
886:
796:
766:
726:
631:
2903:
1135:
746:
3296:
3281:
How to draw a perpendicular bisector of a line with compass and straight edge
3062:
2167:
Now, use the fact that the inner product vanishes for perpendicular vectors:
1491:
1258:
1140:
1125:
1054:
871:
831:
781:
556:
519:
486:
320:
3008:
concerns a property of two perpendicular lines intersecting at a triangle's
1497:
Construction of the perpendicular (blue) to the line AB through the point P.
2860:
2763:
1819:
1724:
1713:
1635:
are parallel, any of the following conclusions leads to all of the others:
1550:
for triangles OPA' and OPB' to conclude that angles POA and POB are equal.
1291:
1224:
1079:
1028:
841:
696:
611:
401:
3009:
2079:
1538:
Step 3 (blue): connect Q and P to construct the desired perpendicular PQ.
1529:
1228:
1115:
988:
806:
741:
669:
641:
616:
26:
1553:
To make the perpendicular to the line g at or through the point P using
2994:
973:
952:
942:
932:
891:
836:
731:
721:
621:
472:
2766:
to that circle at the point where the diameter intersects the circle.
1528:
with center at P to create points A' and B' on the line AB, which are
16:
Relationship between two lines that meet at a right angle (90 degrees)
3058:
3051:
3043:
2921:
2907:
2829:
is the distance from the center point to the point of intersection).
1284:
983:
701:
664:
528:
500:
2882:
is perpendicular to the line from that point through the parabola's
3077:
3066:
3027:
2979:
2872:
2755:
1694:
1686:
1516:
To make the perpendicular to the line AB through the point P using
1212:
1064:
1023:
993:
881:
876:
826:
551:
510:
458:
352:
315:
61:
3085:
3023:
2849:
1276:
on one side of the first line is cut by the second line into two
998:
711:
505:
449:
249:
3261:
1612:, any two lines that are both perpendicular to a third line are
3081:
3039:
2759:
1894:
1525:
947:
937:
816:
761:
636:
599:
587:
542:
495:
413:
78:
2990:
of the sides also play a prominent role in triangle geometry.
2910:
is perpendicular to the conjugate axis and to each directrix.
3054:
that has two pairs of adjacent sides that are perpendicular.
2878:
From a point on the tangent line to a parabola's vertex, the
1280:
1232:
1003:
927:
861:
706:
310:
305:
1773:
The concept of perpendicular distance may be generalized to
1412:
means line segment AB is perpendicular to line segment CD.
594:
444:
1777:
orthogonal distance, between more abstract non-geometric
1723:
is measured by a line segment that is perpendicular to a
1716:
from it to the given point is perpendicular to the line.
1422:
Two planes in space are said to be perpendicular if the
2741:
If one slope goes to zero, the other goes to infinity.
2030:, the graphs of the functions will be perpendicular if
1405:{\displaystyle {\overline {AB}}\perp {\overline {CD}}}
3119:
can be pairwise perpendicular, as exemplified by the
3018:
concerns the relationship of line segments through a
2934:
2855:
The major axis of an ellipse is perpendicular to the
2769:
A line segment through a circle's center bisecting a
2721:
2701:
2670:
2643:
2594:
2470:
2323:
2176:
2132:
2096:
2036:
1971:
1906:
1542:
To prove that the PQ is perpendicular to AB, use the
1368:
1336:
1304:
1568:
1481:
3046:, all pairs of adjacent sides are perpendicular. A
2086:can be also used to obtain the same result: First,
2947:
2733:
2707:
2683:
2656:
2629:
2579:
2455:
2308:
2159:
2118:
2068:
2022:
1957:
1639:One of the angles in the diagram is a right angle.
1404:
1354:
1322:
3096:, in an orthodiagonal quadrilateral that is also
3294:
3231:
2924:that are perpendicular to each other. It has an
1727:to the curve at the nearest point on the curve.
1235:that are 90 degrees or Ļ/2 radians wide) at the
3204:, Dover, 2nd edition, 1996: pp. 104ā105, #4ā23.
2744:
1192:
34:, using "perpendicular" as a noun. The point
3243:(2nd ed.), New York: Barnes & Noble
3110:
1577:The arrowhead marks indicate that the lines
3218:29(4), September 1998, p. 331, problem 635.
1712:on that line. That is the point at which a
2734:{\displaystyle \varepsilon \rightarrow 0.}
1604:) are both perpendicular to a third line (
1429:
1199:
1185:
68:
32:the perpendicular from A to the segment CD
3065:is a perpendicular to a side through the
3001:is perpendicular to the triangle's base.
2784:and divides the other chord into lengths
1719:Likewise, the distance from a point to a
1671:
1247:may be represented graphically using the
2630:{\displaystyle \therefore m_{1}m_{2}=-1}
1818:
1697:from one to the other, measured along a
1572:
1520:, proceed as follows (see figure left):
1415:A line is said to be perpendicular to a
25:
3295:
2982:are perpendicular to their respective
1701:that is perpendicular to one or both.
1283:. Perpendicularity can be shown to be
299:Straightedge and compass constructions
3080:are perpendicular. These include the
2023:{\displaystyle y_{2}(x)=m_{2}x+b_{2}}
1958:{\displaystyle y_{1}(x)=m_{1}x+b_{1}}
1814:
1796:, between an arbitrary point and its
1518:compass-and-straightedge construction
1426:at which they meet is a right angle.
1823:Two perpendicular lines have slopes
3247:
3188:
3176:
3164:
2810:equals the square of the diameter.
1800:on the surface. It can be used for
1752:Nearest distance between skew lines
1330:is perpendicular to a line segment
1290:Perpendicularity easily extends to
13:
2880:other tangent line to the parabola
14:
3314:
3268:
3033:
2971:are perpendicular to each other.
1569:In relationship to parallel lines
1482:Construction of the perpendicular
265:Noncommutative algebraic geometry
3202:Challenging Problems in Geometry
3146:Tangential and normal components
1744:Point on plane closest to origin
1732:distance from a point to a plane
1677:This section is an excerpt from
1502:
1490:
1454:is the point of intersection of
1355:{\displaystyle {\overline {CD}}}
1323:{\displaystyle {\overline {AB}}}
2773:is perpendicular to the chord.
1708:is the distance to the nearest
1706:distance from a point to a line
40:foot of the perpendicular from
3207:
3194:
3182:
3170:
3158:
3022:and perpendicular to any line
2957:
2725:
2500:
2478:
2447:
2412:
2387:
2362:
2331:
2300:
2265:
2240:
2215:
2184:
2151:
2133:
2119:{\displaystyle {\vec {r}}_{j}}
2104:
2069:{\displaystyle m_{1}m_{2}=-1.}
1988:
1982:
1923:
1917:
1616:to each other, because of the
1557:, see the animation at right.
1474:of this perpendicular through
1298:. For example, a line segment
658:- / other-dimensional
21:Perpendicular (disambiguation)
1:
3225:
2897:
1869:satisfying the relationship
1792:normal distance, involving a
1787:principal components analysis
3127:axes of a three-dimensional
2962:
2948:{\displaystyle {\sqrt {2}}.}
2866:
2708:{\displaystyle \varepsilon }
1458:and the unique line through
1397:
1379:
1347:
1315:
7:
3277:with interactive animation.
3215:College Mathematics Journal
3134:
3129:Cartesian coordinate system
3074:orthodiagonal quadrilateral
2839:
2825:is the circle's radius and
2745:In circles and other conics
1693:between two objects is the
10:
3319:
3254:Holt, Rinehart and Winston
2749:
2715:, and take the limit that
1676:
1667:All four angles are equal.
1524:Step 1 (red): construct a
18:
3289:(animated demonstration).
3283:(animated demonstration).
3275:Definition: perpendicular
3200:Posamentier and Salkind,
3111:Lines in three dimensions
3076:is a quadrilateral whose
1897:equals ā1. Thus for two
1738:Other instances include:
1660:is perpendicular to line
1649:is perpendicular to line
1462:that is perpendicular to
1239:of intersection called a
3233:Altshiller-Court, Nathan
3151:
2762:is perpendicular to the
2160:{\displaystyle (j=1,2).}
1760:Perpendicular regression
154:Non-Archimedean geometry
3117:three-dimensional space
3006:Droz-Farny line theorem
2988:perpendicular bisectors
1764:geometric curve fitting
1430:Foot of a perpendicular
260:Noncommutative geometry
3248:Kay, David C. (1969),
3069:of the opposite side.
2949:
2735:
2709:
2685:
2658:
2631:
2581:
2457:
2310:
2161:
2120:
2070:
2024:
1959:
1886:
1691:perpendicular distance
1679:Perpendicular distance
1672:In computing distances
1593:
1548:SAS congruence theorem
1544:SSS congruence theorem
1406:
1356:
1324:
228:Discrete/Combinatorial
55:
3141:Orthogonal projection
3115:Up to three lines in
3094:Brahmagupta's theorem
2950:
2918:rectangular hyperbola
2736:
2710:
2686:
2684:{\displaystyle x_{2}}
2659:
2657:{\displaystyle x_{1}}
2632:
2582:
2458:
2311:
2162:
2121:
2071:
2025:
1960:
1822:
1576:
1407:
1357:
1325:
211:Discrete differential
29:
2932:
2844:The major and minor
2719:
2699:
2668:
2641:
2592:
2468:
2321:
2174:
2130:
2094:
2034:
1969:
1904:
1442:More precisely, let
1366:
1334:
1302:
1250:perpendicular symbol
19:For other uses, see
3105:van Aubel's theorem
1768:total least squares
1562:Pythagorean theorem
1243:. The condition of
478:Pythagorean theorem
3026:to the triangle's
3016:Harcourt's theorem
2999:isosceles triangle
2945:
2891:orthoptic property
2731:
2705:
2681:
2654:
2627:
2577:
2453:
2306:
2157:
2116:
2066:
2020:
1955:
1887:
1815:Graph of functions
1618:parallel postulate
1610:Euclidean geometry
1594:
1402:
1352:
1320:
56:
3057:Each of the four
2940:
2503:
2481:
2450:
2415:
2390:
2365:
2334:
2303:
2268:
2243:
2218:
2187:
2107:
2088:shift coordinates
1804:and for defining
1400:
1382:
1350:
1318:
1217:geometric objects
1209:
1208:
1174:
1173:
897:List of geometers
580:Three-dimensional
569:
568:
46:, or simply, the
3310:
3264:
3250:College Geometry
3244:
3242:
3219:
3211:
3205:
3198:
3192:
3186:
3180:
3174:
3168:
3162:
2954:
2952:
2951:
2946:
2941:
2936:
2809:
2740:
2738:
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2732:
2714:
2712:
2711:
2706:
2690:
2688:
2687:
2682:
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2660:
2655:
2653:
2652:
2636:
2634:
2633:
2628:
2617:
2616:
2607:
2606:
2586:
2584:
2583:
2578:
2570:
2569:
2560:
2559:
2550:
2546:
2545:
2544:
2535:
2534:
2511:
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2505:
2504:
2496:
2489:
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2462:
2460:
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2451:
2443:
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2417:
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2408:
2405:
2404:
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2391:
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2220:
2219:
2211:
2208:
2207:
2195:
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2180:
2166:
2164:
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2158:
2125:
2123:
2122:
2117:
2115:
2114:
2109:
2108:
2100:
2075:
2073:
2072:
2067:
2056:
2055:
2046:
2045:
2029:
2027:
2026:
2021:
2019:
2018:
2003:
2002:
1981:
1980:
1964:
1962:
1961:
1956:
1954:
1953:
1938:
1937:
1916:
1915:
1899:linear functions
1884:
1868:
1845:
1587:transversal line
1555:Thales's theorem
1506:
1494:
1477:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1411:
1409:
1408:
1403:
1401:
1396:
1388:
1383:
1378:
1370:
1361:
1359:
1358:
1353:
1351:
1346:
1338:
1329:
1327:
1326:
1321:
1319:
1314:
1306:
1245:perpendicularity
1201:
1194:
1187:
915:
914:
434:
433:
367:Zero-dimensional
72:
58:
57:
3318:
3317:
3313:
3312:
3311:
3309:
3308:
3307:
3293:
3292:
3271:
3228:
3223:
3222:
3212:
3208:
3199:
3195:
3187:
3183:
3175:
3171:
3163:
3159:
3154:
3137:
3113:
3048:right trapezoid
3036:
2965:
2960:
2935:
2933:
2930:
2929:
2904:transverse axis
2900:
2869:
2842:
2833:Thales' theorem
2793:
2752:
2747:
2720:
2717:
2716:
2700:
2697:
2696:
2675:
2671:
2669:
2666:
2665:
2648:
2644:
2642:
2639:
2638:
2612:
2608:
2602:
2598:
2593:
2590:
2589:
2565:
2561:
2555:
2551:
2540:
2536:
2530:
2526:
2519:
2515:
2506:
2495:
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2484:
2473:
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2469:
2466:
2465:
2442:
2441:
2435:
2431:
2425:
2421:
2407:
2406:
2400:
2396:
2382:
2381:
2375:
2371:
2357:
2356:
2350:
2346:
2337:
2326:
2325:
2324:
2322:
2319:
2318:
2295:
2294:
2288:
2284:
2278:
2274:
2260:
2259:
2253:
2249:
2235:
2234:
2228:
2224:
2210:
2209:
2203:
2199:
2190:
2179:
2178:
2177:
2175:
2172:
2171:
2131:
2128:
2127:
2110:
2099:
2098:
2097:
2095:
2092:
2091:
2051:
2047:
2041:
2037:
2035:
2032:
2031:
2014:
2010:
1998:
1994:
1976:
1972:
1970:
1967:
1966:
1949:
1945:
1933:
1929:
1911:
1907:
1905:
1902:
1901:
1882:
1876:
1870:
1867:
1860:
1853:
1847:
1844:
1837:
1830:
1824:
1817:
1812:
1811:
1806:offset surfaces
1802:surface fitting
1781:objects, as in
1682:
1674:
1625:vertical angles
1592:, are parallel.
1571:
1514:
1513:
1512:
1511:
1510:
1507:
1499:
1498:
1495:
1484:
1475:
1467:
1463:
1459:
1455:
1451:
1447:
1446:be a point and
1443:
1432:
1389:
1387:
1371:
1369:
1367:
1364:
1363:
1339:
1337:
1335:
1332:
1331:
1307:
1305:
1303:
1300:
1299:
1205:
1176:
1175:
912:
911:
902:
901:
692:
691:
675:
674:
660:
659:
647:
646:
583:
582:
571:
570:
431:
430:
428:Two-dimensional
419:
418:
392:
391:
389:One-dimensional
380:
379:
370:
369:
358:
357:
291:
290:
289:
272:
271:
120:
119:
108:
85:
24:
17:
12:
11:
5:
3316:
3306:
3305:
3291:
3290:
3284:
3278:
3270:
3269:External links
3267:
3266:
3265:
3245:
3227:
3224:
3221:
3220:
3206:
3193:
3181:
3169:
3167:, p. 114)
3156:
3155:
3153:
3150:
3149:
3148:
3143:
3136:
3133:
3112:
3109:
3035:
3034:Quadrilaterals
3032:
2969:right triangle
2967:The legs of a
2964:
2961:
2959:
2956:
2944:
2939:
2899:
2896:
2868:
2865:
2841:
2838:
2751:
2748:
2746:
2743:
2730:
2727:
2724:
2704:
2693:
2692:
2678:
2674:
2651:
2647:
2626:
2623:
2620:
2615:
2611:
2605:
2601:
2597:
2587:
2576:
2573:
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2564:
2558:
2554:
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2529:
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2522:
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2449:
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2370:
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2299:
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2256:
2252:
2248:
2242:
2239:
2231:
2227:
2223:
2217:
2214:
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2202:
2198:
2193:
2186:
2183:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2113:
2106:
2103:
2065:
2062:
2059:
2054:
2050:
2044:
2040:
2017:
2013:
2009:
2006:
2001:
1997:
1993:
1990:
1987:
1984:
1979:
1975:
1952:
1948:
1944:
1941:
1936:
1932:
1928:
1925:
1922:
1919:
1914:
1910:
1880:
1874:
1865:
1858:
1851:
1842:
1835:
1828:
1816:
1813:
1810:
1809:
1794:surface normal
1790:
1783:linear algebra
1757:
1756:
1748:
1683:
1675:
1673:
1670:
1669:
1668:
1665:
1654:
1643:
1640:
1596:If two lines (
1570:
1567:
1540:
1539:
1536:
1533:
1508:
1501:
1500:
1496:
1489:
1488:
1487:
1486:
1485:
1483:
1480:
1470:is called the
1431:
1428:
1424:dihedral angle
1399:
1395:
1392:
1386:
1381:
1377:
1374:
1349:
1345:
1342:
1317:
1313:
1310:
1274:straight angle
1207:
1206:
1204:
1203:
1196:
1189:
1181:
1178:
1177:
1172:
1171:
1170:
1169:
1164:
1156:
1155:
1151:
1150:
1149:
1148:
1143:
1138:
1133:
1128:
1123:
1118:
1113:
1108:
1103:
1098:
1090:
1089:
1085:
1084:
1083:
1082:
1077:
1072:
1067:
1062:
1057:
1052:
1047:
1039:
1038:
1034:
1033:
1032:
1031:
1026:
1021:
1016:
1011:
1006:
1001:
996:
991:
986:
981:
976:
968:
967:
963:
962:
961:
960:
955:
950:
945:
940:
935:
930:
922:
921:
913:
909:
908:
907:
904:
903:
900:
899:
894:
889:
884:
879:
874:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
693:
689:
688:
687:
684:
683:
677:
676:
673:
672:
667:
661:
654:
653:
652:
649:
648:
645:
644:
639:
634:
632:Platonic Solid
629:
624:
619:
614:
609:
604:
603:
602:
591:
590:
584:
578:
577:
576:
573:
572:
567:
566:
565:
564:
559:
554:
546:
545:
539:
538:
537:
536:
531:
523:
522:
516:
515:
514:
513:
508:
503:
498:
490:
489:
483:
482:
481:
480:
475:
470:
462:
461:
455:
454:
453:
452:
447:
442:
432:
426:
425:
424:
421:
420:
417:
416:
411:
410:
409:
404:
393:
387:
386:
385:
382:
381:
378:
377:
371:
365:
364:
363:
360:
359:
356:
355:
350:
345:
339:
338:
333:
328:
318:
313:
308:
302:
301:
292:
288:
287:
284:
280:
279:
278:
277:
274:
273:
270:
269:
268:
267:
257:
252:
247:
242:
237:
236:
235:
225:
220:
215:
214:
213:
208:
203:
193:
192:
191:
186:
176:
171:
166:
161:
156:
151:
150:
149:
144:
143:
142:
127:
121:
115:
114:
113:
110:
109:
107:
106:
96:
90:
87:
86:
73:
65:
64:
38:is called the
15:
9:
6:
4:
3:
2:
3315:
3304:
3303:Orthogonality
3301:
3300:
3298:
3288:
3285:
3282:
3279:
3276:
3273:
3272:
3263:
3259:
3255:
3251:
3246:
3241:
3240:
3234:
3230:
3229:
3217:
3216:
3210:
3203:
3197:
3191:, p. 91)
3190:
3185:
3179:, p. 91)
3178:
3173:
3166:
3161:
3157:
3147:
3144:
3142:
3139:
3138:
3132:
3130:
3126:
3122:
3118:
3108:
3106:
3101:
3099:
3095:
3091:
3087:
3083:
3079:
3075:
3070:
3068:
3064:
3063:quadrilateral
3060:
3055:
3053:
3049:
3045:
3041:
3031:
3029:
3025:
3021:
3017:
3013:
3011:
3007:
3002:
3000:
2996:
2991:
2989:
2985:
2981:
2977:
2972:
2970:
2955:
2942:
2937:
2927:
2923:
2919:
2914:
2911:
2909:
2905:
2895:
2892:
2887:
2885:
2881:
2876:
2874:
2864:
2862:
2858:
2853:
2851:
2847:
2837:
2834:
2830:
2828:
2824:
2820:
2816:
2811:
2808:
2804:
2800:
2796:
2791:
2787:
2783:
2779:
2774:
2772:
2767:
2765:
2761:
2757:
2742:
2728:
2722:
2702:
2676:
2672:
2649:
2645:
2624:
2621:
2618:
2613:
2609:
2603:
2599:
2595:
2588:
2574:
2571:
2566:
2562:
2556:
2552:
2547:
2541:
2537:
2531:
2527:
2523:
2520:
2516:
2512:
2507:
2497:
2490:
2485:
2475:
2464:
2444:
2436:
2432:
2426:
2422:
2418:
2409:
2401:
2397:
2393:
2384:
2376:
2372:
2368:
2359:
2351:
2347:
2343:
2338:
2328:
2317:
2297:
2289:
2285:
2279:
2275:
2271:
2262:
2254:
2250:
2246:
2237:
2229:
2225:
2221:
2212:
2204:
2200:
2196:
2191:
2181:
2170:
2169:
2168:
2154:
2148:
2145:
2142:
2139:
2136:
2111:
2101:
2089:
2085:
2081:
2076:
2063:
2060:
2057:
2052:
2048:
2042:
2038:
2015:
2011:
2007:
2004:
1999:
1995:
1991:
1985:
1977:
1973:
1950:
1946:
1942:
1939:
1934:
1930:
1926:
1920:
1912:
1908:
1900:
1896:
1892:
1879:
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1850:
1841:
1834:
1827:
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1807:
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1774:
1771:
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1700:
1696:
1692:
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1634:
1630:
1626:
1621:
1619:
1615:
1611:
1607:
1603:
1599:
1591:
1588:
1585:, cut by the
1584:
1580:
1575:
1566:
1563:
1558:
1556:
1551:
1549:
1545:
1537:
1534:
1531:
1527:
1523:
1522:
1521:
1519:
1505:
1493:
1479:
1473:
1440:
1437:
1427:
1425:
1420:
1418:
1413:
1393:
1390:
1384:
1375:
1372:
1343:
1340:
1311:
1308:
1297:
1293:
1288:
1286:
1282:
1279:
1275:
1269:
1267:
1266:
1265:normal vector
1261:
1260:
1259:orthogonality
1254:
1252:
1251:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1221:perpendicular
1218:
1214:
1202:
1197:
1195:
1190:
1188:
1183:
1182:
1180:
1179:
1168:
1165:
1163:
1160:
1159:
1158:
1157:
1153:
1152:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1093:
1092:
1091:
1087:
1086:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1042:
1041:
1040:
1036:
1035:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
975:
972:
971:
970:
969:
965:
964:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
925:
924:
923:
920:
917:
916:
906:
905:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
694:
686:
685:
682:
679:
678:
671:
668:
666:
663:
662:
657:
651:
650:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
608:
605:
601:
598:
597:
596:
593:
592:
589:
586:
585:
581:
575:
574:
563:
560:
558:
557:Circumference
555:
553:
550:
549:
548:
547:
544:
541:
540:
535:
532:
530:
527:
526:
525:
524:
521:
520:Quadrilateral
518:
517:
512:
509:
507:
504:
502:
499:
497:
494:
493:
492:
491:
488:
487:Parallelogram
485:
484:
479:
476:
474:
471:
469:
466:
465:
464:
463:
460:
457:
456:
451:
448:
446:
443:
441:
438:
437:
436:
435:
429:
423:
422:
415:
412:
408:
405:
403:
400:
399:
398:
395:
394:
390:
384:
383:
376:
373:
372:
368:
362:
361:
354:
351:
349:
346:
344:
341:
340:
337:
334:
332:
329:
326:
325:Perpendicular
322:
321:Orthogonality
319:
317:
314:
312:
309:
307:
304:
303:
300:
297:
296:
295:
285:
282:
281:
276:
275:
266:
263:
262:
261:
258:
256:
253:
251:
248:
246:
245:Computational
243:
241:
238:
234:
231:
230:
229:
226:
224:
221:
219:
216:
212:
209:
207:
204:
202:
199:
198:
197:
194:
190:
187:
185:
182:
181:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
148:
145:
141:
138:
137:
136:
133:
132:
131:
130:Non-Euclidean
128:
126:
123:
122:
118:
112:
111:
104:
100:
97:
95:
92:
91:
89:
88:
84:
80:
76:
71:
67:
66:
63:
60:
59:
53:
49:
45:
44:to segment CD
41:
37:
33:
28:
22:
3252:, New York:
3249:
3238:
3213:
3209:
3201:
3196:
3184:
3172:
3160:
3124:
3120:
3114:
3102:
3071:
3056:
3037:
3014:
3003:
2992:
2973:
2966:
2926:eccentricity
2915:
2912:
2901:
2888:
2877:
2870:
2861:latus rectum
2859:and to each
2854:
2843:
2831:
2826:
2822:
2818:
2814:
2812:
2806:
2802:
2798:
2794:
2789:
2785:
2781:
2777:
2775:
2768:
2764:tangent line
2753:
2694:
2077:
1888:
1877:
1871:
1862:
1855:
1848:
1839:
1832:
1825:
1772:
1758:
1750:
1742:
1737:
1729:
1725:tangent line
1718:
1703:
1684:
1661:
1657:
1650:
1646:
1632:
1628:
1622:
1605:
1601:
1597:
1595:
1589:
1582:
1578:
1559:
1552:
1541:
1515:
1471:
1441:
1435:
1433:
1421:
1414:
1289:
1270:
1263:
1257:
1255:
1248:
1244:
1240:
1229:right angles
1225:intersection
1220:
1210:
1029:Parameshvara
842:Parameshvara
612:Dodecahedron
324:
196:Differential
51:
47:
43:
39:
35:
31:
3010:orthocenter
2958:In polygons
2080:dot product
1530:equidistant
1450:a line. If
1154:Present day
1101:Lobachevsky
1088:1700sā1900s
1045:Jyeį¹£į¹hadeva
1037:1400sā1700s
989:Brahmagupta
812:Lobachevsky
792:Jyeį¹£į¹hadeva
742:Brahmagupta
670:Hypersphere
642:Tetrahedron
617:Icosahedron
189:Diophantine
3226:References
3088:, and the
3059:maltitudes
2995:Euler line
2922:asymptotes
2898:Hyperbolas
2691:vanishes.)
1883:= −1
1779:orthogonal
1014:al-Yasamin
958:Apollonius
953:Archimedes
943:Pythagoras
933:Baudhayana
887:al-Yasamin
837:Pythagoras
732:Baudhayana
722:Archimedes
717:Apollonius
622:Octahedron
473:Hypotenuse
348:Similarity
343:Congruence
255:Incidence
206:Symplectic
201:Riemannian
184:Arithmetic
159:Projective
147:Hyperbolic
75:Projecting
3235:(1952) ,
3189:Kay (1969
3177:Kay (1969
3165:Kay (1969
3078:diagonals
3052:trapezoid
3044:rectangle
3042:or other
2976:altitudes
2963:Triangles
2928:equal to
2908:hyperbola
2867:Parabolas
2857:directrix
2726:→
2723:ε
2703:ε
2637:(unless
2622:−
2596:∴
2501:→
2491:⋅
2479:→
2448:^
2413:^
2388:^
2363:^
2332:→
2301:^
2266:^
2241:^
2216:^
2185:→
2105:→
2061:−
1893:of their
1434:The word
1398:¯
1385:⊥
1380:¯
1348:¯
1316:¯
1285:symmetric
1278:congruent
1223:if their
1131:Minkowski
1050:Descartes
984:Aryabhata
979:KÄtyÄyana
910:by period
822:Minkowski
797:KÄtyÄyana
757:Descartes
702:Aryabhata
681:Geometers
665:Tesseract
529:Trapezoid
501:Rectangle
294:Dimension
179:Algebraic
169:Synthetic
140:Spherical
125:Euclidean
3297:Category
3262:69-12075
3135:See also
3067:midpoint
3028:incircle
2980:triangle
2873:parabola
2840:Ellipses
2756:diameter
1695:distance
1687:geometry
1614:parallel
1292:segments
1213:geometry
1121:PoincarƩ
1065:Minggatu
1024:Yang Hui
994:Virasena
882:Yang Hui
877:Virasena
847:PoincarƩ
827:Minggatu
607:Cylinder
552:Diameter
511:Rhomboid
468:Altitude
459:Triangle
353:Symmetry
331:Parallel
316:Diagonal
286:Features
283:Concepts
174:Analytic
135:Elliptic
117:Branches
103:Timeline
62:Geometry
48:foot of
3086:rhombus
3024:tangent
2850:ellipse
2821:(where
2792:, then
2750:Circles
2084:vectors
1891:product
1785:(e.g.,
1714:segment
1532:from P.
1466:, then
1146:Coxeter
1126:Hilbert
1111:Riemann
1060:Huygens
1019:al-Tusi
1009:KhayyƔm
999:Alhazen
966:1ā1400s
867:al-Tusi
852:Riemann
802:KhayyƔm
787:Huygens
782:Hilbert
752:Coxeter
712:Alhazen
690:by name
627:Pyramid
506:Rhombus
450:Polygon
402:segment
250:Fractal
233:Digital
218:Complex
99:History
94:Outline
3260:
3123:, and
3098:cyclic
3084:, the
3082:square
3040:square
3020:vertex
2997:of an
2986:. The
2848:of an
2760:circle
2126:, for
1895:slopes
1689:, the
1526:circle
1281:angles
1233:angles
1227:forms
1215:, two
1167:Gromov
1162:Atiyah
1141:Veblen
1136:Cartan
1106:Bolyai
1075:Sakabe
1055:Pascal
948:Euclid
938:Manava
872:Veblen
857:Sakabe
832:Pascal
817:Manava
777:Gromov
762:Euclid
747:Cartan
737:Bolyai
727:Atiyah
637:Sphere
600:cuboid
588:Volume
543:Circle
496:Square
414:Length
336:Vertex
240:Convex
223:Finite
164:Affine
79:sphere
3152:Notes
3092:. By
3061:of a
3050:is a
3038:In a
2984:bases
2978:of a
2906:of a
2884:focus
2871:In a
2771:chord
2758:of a
2754:Each
1846:and
1721:curve
1710:point
1656:Line
1645:Line
1417:plane
1237:point
1116:Klein
1096:Gauss
1070:Euler
1004:Sijzi
974:Zhang
928:Ahmes
892:Zhang
862:Sijzi
807:Klein
772:Gauss
767:Euler
707:Ahmes
440:Plane
375:Point
311:Curve
306:Angle
83:plane
81:to a
52:on CD
3258:LCCN
3121:x, y
3090:kite
3004:The
2993:The
2974:The
2920:has
2902:The
2889:The
2846:axes
2788:and
2780:and
2078:The
1965:and
1798:foot
1730:The
1704:The
1699:line
1631:and
1600:and
1581:and
1560:The
1472:foot
1436:foot
1296:rays
1294:and
1241:foot
1219:are
1080:Aida
697:Aida
656:Four
595:Cube
562:Area
534:Kite
445:Area
397:Line
3103:By
3072:An
2817:ā 4
2664:or
2082:of
1854:= Ī
1831:= Ī
1685:In
1211:In
919:BCE
407:ray
3299::
3256:,
3131:.
3030:.
3012:.
2916:A
2886:.
2863:.
2805:+
2801:+
2797:+
2729:0.
2064:1.
1861:/Ī
1838:/Ī
1789:);
1770:.
1478:.
1268:.
77:a
54:.
3125:z
2943:.
2938:2
2827:p
2823:r
2819:p
2815:r
2807:d
2803:c
2799:b
2795:a
2790:d
2786:c
2782:b
2778:a
2677:2
2673:x
2650:1
2646:x
2625:1
2619:=
2614:2
2610:m
2604:1
2600:m
2575:0
2572:=
2567:2
2563:x
2557:1
2553:x
2548:)
2542:2
2538:m
2532:1
2528:m
2524:+
2521:1
2517:(
2513:=
2508:2
2498:r
2486:1
2476:r
2445:y
2437:2
2433:x
2427:2
2423:m
2419:+
2410:x
2402:2
2398:x
2394:=
2385:y
2377:2
2373:y
2369:+
2360:x
2352:2
2348:x
2344:=
2339:2
2329:r
2298:y
2290:1
2286:x
2280:1
2276:m
2272:+
2263:x
2255:1
2251:x
2247:=
2238:y
2230:1
2226:y
2222:+
2213:x
2205:1
2201:x
2197:=
2192:1
2182:r
2155:.
2152:)
2149:2
2146:,
2143:1
2140:=
2137:j
2134:(
2112:j
2102:r
2058:=
2053:2
2049:m
2043:1
2039:m
2016:2
2012:b
2008:+
2005:x
2000:2
1996:m
1992:=
1989:)
1986:x
1983:(
1978:2
1974:y
1951:1
1947:b
1943:+
1940:x
1935:1
1931:m
1927:=
1924:)
1921:x
1918:(
1913:1
1909:y
1885:.
1881:2
1878:m
1875:1
1872:m
1866:2
1863:x
1859:2
1856:y
1852:2
1849:m
1843:1
1840:x
1836:1
1833:y
1829:1
1826:m
1808:.
1681:.
1664:.
1662:b
1658:c
1653:.
1651:a
1647:c
1633:b
1629:a
1606:c
1602:b
1598:a
1590:c
1583:b
1579:a
1476:A
1468:B
1464:m
1460:A
1456:m
1452:B
1448:m
1444:A
1394:D
1391:C
1376:B
1373:A
1344:D
1341:C
1312:B
1309:A
1231:(
1200:e
1193:t
1186:v
327:)
323:(
105:)
101:(
50:A
42:A
36:B
23:.
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