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of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two paths each of whose length equals the semiperimeter. The three cleavers concur at the
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touches the triangle partition the triangle's perimeter into two equal lengths, thus creating two paths each of which has a length equal to the semiperimeter. If
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1139:) pairs of opposite sides have lengths summing to the semiperimeter—namely, the area is the product of the inradius and the semiperimeter:
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and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter
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equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is
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In any triangle, the distance along the boundary of the triangle from a vertex to the point on the opposite edge touched by an
449:{\displaystyle {\begin{aligned}s&=|AB|+|A'B|=|AB|+|AB'|=|AC|+|A'C|\\&=|AC|+|AC'|=|BC|+|B'C|=|BC|+|BC'|.\end{aligned}}}
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35:. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for
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The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths
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1447:{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}},}
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of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius.
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are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (
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1287:{\displaystyle K={\sqrt {\left(s-a\right)\left(s-b\right)\left(s-c\right)\left(s-d\right)}}.}
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a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius
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The area of a triangle can also be calculated from its semiperimeter and side lengths
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One of the triangle area formulas involving the semiperimeter also applies to
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of a triangle can also be calculated from the semiperimeter and side lengths:
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has a form similar to that of Heron's formula for the triangle area:
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779:{\displaystyle R={\frac {abc}{4{\sqrt {s(s-a)(s-b)(s-c)}}}}.}
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In any triangle, any vertex and the point where the opposite
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980:{\displaystyle t_{a}={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}.}
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874:{\displaystyle r={\sqrt {\frac {(s-a)(s-b)(s-c)}{s}}}.}
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A triangle's semiperimeter equals the perimeter of its
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the perimeter if and only if it also bisects the area.
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16:Half of the sum of side lengths of a polygon
491:of all the points on the triangle's edges.
1603:. Mineola, New York: Dover. p. 70.
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1121:{\displaystyle s={\frac {a+b+c+d}{2}}.}
1061:The formula for the semiperimeter of a
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532:of any triangle is the product of its
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789:This formula can be derived from the
112:{\displaystyle s={\frac {a+b+c}{2}}.}
519:Formulas involving the semiperimeter
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1601:Advanced Euclidean Geometry
1573:
10:
1673:
1599:Johnson, Roger A. (2007).
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1472:are the four solutions of
1042:{\displaystyle (s-a)(s-b)}
1465:are two opposite angles.
1133:tangential quadrilaterals
59:equals the semiperimeter.
1300:generalizes this to all
1503:The semiperimeter of a
1470:bicentric quadrilateral
1298:Bretschneider's formula
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564:{\displaystyle A=rs.}
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47:Motivation: triangles
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1468:The four sides of a
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513:triangle inequality
1630:Weisstein, Eric W.
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1181:for the area of a
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1633:"Semiperimeter"
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1580:Semidiameter
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133:A, B, B', C'
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31:is half its
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465:Nagel point
1586:References
1067:a, b, c, d
1000:hypotenuse
890:cotangents
888:gives the
123:Properties
1638:MathWorld
1540:⋅
1537:π
1457:in which
1427:γ
1421:α
1411:
1398:⋅
1383:−
1374:−
1359:−
1344:−
1329:−
1269:−
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760:−
745:−
730:−
650:−
631:−
612:−
528:The area
151:splitters
37:triangles
33:perimeter
1651:Category
1574:See also
996:excircle
534:inradius
496:incenter
481:incircle
431:′
385:′
345:′
292:′
252:′
206:′
129:excircle
57:excircle
21:geometry
1499:Circles
1493:apothem
998:on the
576:a, b, c
511:By the
499:bisects
483:of the
472:cleaver
463:at the
65:a, b, c
29:polygon
1607:
1513:radius
1505:circle
1486:convex
1302:convex
1049:where
578:using
461:concur
153:, and
23:, the
990:In a
27:of a
1605:ISBN
1461:and
1051:a, b
884:The
678:The
1402:cos
1069:is
903:is
146:CC'
142:BB'
138:AA'
19:In
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1564:pi
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1028:s
1025:(
1022:)
1019:a
1013:s
1010:(
975:.
969:c
966:+
963:b
956:)
953:a
947:s
944:(
941:s
938:c
935:b
930:2
924:=
919:a
915:t
901:a
869:.
863:s
859:)
856:c
850:s
847:(
844:)
841:b
835:s
832:(
829:)
826:a
820:s
817:(
810:=
807:r
774:.
766:)
763:c
757:s
754:(
751:)
748:b
742:s
739:(
736:)
733:a
727:s
724:(
721:s
716:4
711:c
708:b
705:a
699:=
696:R
683:R
663:.
657:)
653:c
647:s
643:(
638:)
634:b
628:s
624:(
619:)
615:a
609:s
605:(
601:s
596:=
593:A
559:.
556:s
553:r
550:=
547:A
530:A
440:.
436:|
428:C
424:B
420:|
416:+
412:|
408:C
405:B
401:|
397:=
393:|
389:C
382:B
377:|
373:+
369:|
365:C
362:B
358:|
354:=
350:|
342:C
338:A
334:|
330:+
326:|
322:C
319:A
315:|
311:=
300:|
296:C
289:A
284:|
280:+
276:|
272:C
269:A
265:|
261:=
257:|
249:B
245:A
241:|
237:+
233:|
229:B
226:A
222:|
218:=
214:|
210:B
203:A
198:|
194:+
190:|
186:B
183:A
179:|
175:=
168:s
107:.
102:2
98:c
95:+
92:b
89:+
86:a
80:=
77:s
41:s
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