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1376:{\displaystyle {\begin{alignedat}{3}E'&=-{\frac {1}{a}}&&:{\phantom {-}}{\frac {1}{b}}&&:{\phantom {-}}{\frac {1}{c}},\\F'&={\phantom {-}}{\frac {1}{a}}&&:-{\frac {1}{b}}&&:{\phantom {-}}{\frac {1}{c}},\\D'&={\phantom {-}}{\frac {1}{a}}&&:{\phantom {-}}{\frac {1}{b}}&&:-{\frac {1}{c}}.\end{alignedat}}}
485:
670:{\displaystyle {\begin{alignedat}{3}E&={}\,0&&:{\frac {1}{b}}&&:{\frac {1}{c}},\\F&={}{\frac {1}{a}}&&:\,0&&:{\frac {1}{c}},\\D&={}{\frac {1}{a}}&&:{\frac {1}{b}}&&:\,0.\end{alignedat}}}
106:). In general, a midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to half the length of the third side.
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with ratio -1/2. Thus, the sides of the medial triangle are half and parallel to the corresponding sides of triangle ABC. Hence, the medial triangle is inversely
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The red triangle is the medial triangle of the black. The endpoints of the red triangle coincide with the midpoints of the black triangle.
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The name "anticomplementary triangle" corresponds to the fact that its vertices are the anticomplements of the vertices
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circumscribes the medial triangle, and so the nine-point center is the circumcenter of the medial triangle.
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238:, so their areas are equal and thus the area of each is 1/4 the area of the original triangle.
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of the reference triangle. The vertices of the medial triangle are the complements of
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of the triangle if and only if the point lies in the interior of the medial triangle.
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1531:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.
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to the triangle whose vertices are the midpoints between the reference triangle's
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Franzsen, William N.. "The distance from the incenter to the Euler line",
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of the circumcenter, centroid and orthocenter. The medial triangle is the
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for which none of the other three interior triangles has smaller area.
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Triangle defined from the midpoints of the sides of another triangle
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Torrejon, Ricardo M. "On an Erdos inscribed triangle inequality",
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84:, which is the triangle whose sides have the same lengths as the
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The medial triangle can also be viewed as the image of triangle
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Chakerian, G. D. "A Distorted View of
Geometry." Ch. 7 in
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A point in the interior of a triangle is the center of an
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http://forumgeom.fau.edu/FG2005volume5/FG200519index.html
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sides. The medial triangle is not the same thing as the
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98:Each side of the medial triangle is called a
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256:. This fact provides a tool for proving
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204:. It also follows from this that the
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408:{\displaystyle a=|BC|,b=|CA|,c=|AB|}
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193:and shares the same centroid and
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307:The medial triangle is the only
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808:{\displaystyle \triangle EFD.}
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440:{\displaystyle \triangle ABC.}
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275:of the medial triangle is the
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1085:{\displaystyle \triangle ABC}
867:{\displaystyle \triangle ABC}
837:{\displaystyle \triangle ABC}
768:{\displaystyle \triangle ABC}
707:{\displaystyle \triangle EFD}
472:{\displaystyle \triangle EFD}
109:
1585:"Anticomplementary Triangle"
7:
1503:. Dover Publications, 2007.
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279:of its reference triangle.
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1499:Altshiller-Court, Nathan.
777:anticomplementary triangle
714:is the medial triangle of
681:Anticomplementary triangle
1487:, Prometheus Books, 2012.
264:of the circumcenter. The
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57:of the triangle's sides
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1446:{\displaystyle A,B,C.}
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1609:Elementary geometry
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1517:11 (2011): 231–236.
1515:Forum Geometricorum
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781:antimedial triangle
1582:Weisstein, Eric W.
1563:Weisstein, Eric W.
1544:5, 2005, 137–141.
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1007:{\displaystyle A.}
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121:: circumcenter of
29:Euclidean geometry
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247:circumcenter
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68:case of the
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61:. It is the
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322:Coordinates
288:orthocenter
273:Nagel point
243:orthocenter
1603:Categories
1469:References
110:Properties
100:midsegment
59:AB, AC, BC
1590:MathWorld
1571:MathWorld
1354:−
1330:−
1303:−
1262:−
1238:−
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1071:△
1027:△
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823:△
791:△
754:△
722:△
693:△
458:△
423:△
302:inellipse
284:congruent
232:congruent
206:perimeter
183:homothety
55:midpoints
1456:See also
1289:′
1200:′
1111:′
1050:′
1042:′
1034:′
991:through
945:through
899:through
295:incenter
277:incenter
187:centroid
51:vertices
41:triangle
775:is the
195:medians
191:similar
104:midline
86:medians
74:polygon
53:at the
31:, the
746:then
76:with
72:of a
39:of a
326:Let
293:The
271:The
241:The
221:area
164:and
102:(or
783:of
779:or
685:If
253:ABC
236:SSS
234:by
227:ABC
216:ABC
201:ABC
178:ABC
168:DEF
161:ABC
150:DEF
143:ABC
132:DEF
125:ABC
92:ABC
88:of
66:= 3
46:ABC
35:or
27:In
1605::
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661:0.
318:.
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1367:.
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1297:=
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1197:F
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1167::
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1140::
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1127:1
1119:=
1108:E
1080:C
1077:B
1074:A
1047:D
1039:F
1031:E
1002:.
999:A
979:C
976:B
956:,
953:B
933:C
930:A
910:,
907:C
887:B
884:A
874::
862:C
859:B
856:A
832:C
829:B
826:A
803:.
800:D
797:F
794:E
763:C
760:B
757:A
734:,
731:C
728:B
725:A
702:D
699:F
696:E
657::
647:b
644:1
639::
629:a
626:1
619:=
612:D
605:,
600:c
597:1
592::
584:0
580::
570:a
567:1
560:=
553:F
546:,
541:c
538:1
533::
523:b
520:1
515::
507:0
501:=
494:E
467:D
464:F
461:E
435:.
432:C
429:B
426:A
402:|
398:B
395:A
391:|
387:=
384:c
381:,
377:|
373:A
370:C
366:|
362:=
359:b
356:,
352:|
348:C
345:B
341:|
337:=
334:a
251:△
225:△
214:△
199:△
176:△
166:△
159:△
155:S
148:△
141:△
137:N
130:△
123:△
119:M
90:△
78:n
64:n
44:△
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