27:
443:
The converse of the statement above is true as well, that is if the equation holds for the pedal points of three perpendiculars on the three triangle sides then they intersect in a common point. Therefore, the equation provides a necessary and sufficient condition.
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47:
for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection. The theorem can also be thought of as a generalization of the
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138:
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and therefore the equation above holds. which means all three perpendicular bisectors intersect in the same point.
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Mathematisch für fortgeschrittene Anfänger : Weitere beliebte Beiträge von
Matroids Matheplanet
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consider three lines that are perpendicular to the triangle sides and intersect in a common point
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Carnot's theorem: if three perpendiculars on triangle sides intersect in a common point F, then
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of a triangle to intersect in a common point. In the case of perpendicular bisectors you have
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433:{\displaystyle |AP_{c}|^{2}+|BP_{a}|^{2}+|CP_{b}|^{2}=|BP_{c}|^{2}+|CP_{a}|^{2}+|AP_{b}|^{2}}
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Condition for 3 lines with common point to be perpendicular to the sides of triangle
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501:, then we can construct three perpendiculars on the sides that intersect in
1274:(in German). Heidelberg: Spektrum Akademischer Verlag. pp. 273–276.
1350:
1002:. The equation of Carnot's Theorem then yields the Pythagorean theorem
25:
193:
are the pedal points of those three perpendiculars on the sides
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1335:Vergessene Sätze am Dreieck - Der Satz von Carnot
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225:, then the following equation holds:
1058:Another corollary is the property of
7:
1304:. New York: Dover. pp. 85–86.
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66:
45:necessary and sufficient condition
14:
1270:Wohlgemuth, Martin., ed. (2010).
1253:{\displaystyle |CP_{b}|=|AP_{b}|}
1187:{\displaystyle |BP_{a}|=|CP_{a}|}
1121:{\displaystyle |AP_{c}|=|BP_{c}|}
1048:{\displaystyle a^{2}+b^{2}=c^{2}}
186:{\displaystyle P_{a},P_{b},P_{c}}
21:Carnot's theorem (disambiguation)
1302:Challenging Problems in Geometry
587:, and the line perpendicular to
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1:
1300:; Charles T. Salkind (1996).
474:{\displaystyle \triangle ABC}
81:{\displaystyle \triangle ABC}
547:, the line perpendicular to
1337:at matheplanet.com (German)
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995:{\displaystyle |BP_{c}|=c}
949:{\displaystyle |BP_{a}|=a}
903:{\displaystyle |CP_{b}|=b}
857:{\displaystyle |CP_{a}|=0}
811:{\displaystyle |AP_{c}|=0}
765:{\displaystyle |AP_{b}|=0}
18:
1366:Theorems about triangles
1060:perpendicular bisectors
719:{\displaystyle P_{c}=A}
686:{\displaystyle P_{b}=A}
653:{\displaystyle P_{a}=C}
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1298:Alfred S. Posamentier
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481:has a right angle in
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218:{\displaystyle a,b,c}
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113:{\displaystyle a,b,c}
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1351:Interactive Geometry
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32:blue area = red area
19:For other uses, see
1343:at cut-the-knot.org
520:{\displaystyle F=A}
49:Pythagorean theorem
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620:{\displaystyle A}
600:{\displaystyle c}
580:{\displaystyle A}
560:{\displaystyle b}
540:{\displaystyle b}
494:{\displaystyle C}
452:If the triangle
133:{\displaystyle F}
1373:
1347:Carnot's theorem
1341:Carnot's theorem
1333:Florian Modler:
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37:Carnot's theorem
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627:. Then we have
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59:For a triangle
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1328:External links
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448:Special cases
447:
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405:
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165:
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127:
107:
104:
101:
98:
95:
75:
72:
69:
54:
52:
50:
46:
42:
41:Lazare Carnot
39:(named after
38:
28:
22:
1301:
1271:
1057:
451:
442:
58:
36:
35:
527:: the side
88:with sides
1264:References
1320:829151719
1290:699828882
726:and thus
460:△
67:△
1360:Category
55:Theorem
1318:
1308:
1288:
1278:
140:. If
1316:OCLC
1306:ISBN
1286:OCLC
1276:ISBN
1194:and
956:and
693:and
1349:at
1362::
1314:.
1284:.
1128:,
1055:.
910:,
864:,
818:,
772:,
660:,
51:.
1322:.
1292:.
1247:|
1241:b
1237:P
1233:A
1229:|
1225:=
1221:|
1215:b
1211:P
1207:C
1203:|
1181:|
1175:a
1171:P
1167:C
1163:|
1159:=
1155:|
1149:a
1145:P
1141:B
1137:|
1115:|
1109:c
1105:P
1101:B
1097:|
1093:=
1089:|
1083:c
1079:P
1075:A
1071:|
1041:2
1037:c
1033:=
1028:2
1024:b
1020:+
1015:2
1011:a
990:c
987:=
983:|
977:c
973:P
969:B
965:|
944:a
941:=
937:|
931:a
927:P
923:B
919:|
898:b
895:=
891:|
885:b
881:P
877:C
873:|
852:0
849:=
845:|
839:a
835:P
831:C
827:|
806:0
803:=
799:|
793:c
789:P
785:A
781:|
760:0
757:=
753:|
747:b
743:P
739:A
735:|
714:A
711:=
706:c
702:P
681:A
678:=
673:b
669:P
648:C
645:=
640:a
636:P
615:A
595:c
575:A
555:b
535:b
515:A
512:=
509:F
489:C
469:C
466:B
463:A
426:2
421:|
414:b
410:P
406:A
402:|
398:+
393:2
388:|
381:a
377:P
373:C
369:|
365:+
360:2
355:|
348:c
344:P
340:B
336:|
332:=
327:2
322:|
315:b
311:P
307:C
303:|
299:+
294:2
289:|
282:a
278:P
274:B
270:|
266:+
261:2
256:|
249:c
245:P
241:A
237:|
213:c
210:,
207:b
204:,
201:a
179:c
175:P
171:,
166:b
162:P
158:,
153:a
149:P
128:F
108:c
105:,
102:b
99:,
96:a
76:C
73:B
70:A
23:.
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