Talk:三角形的五心

外心基本性質(莊坤霖)

ABOD,BCOE

(1)OA=OB=OC

(2)若AF=FCOFAC
AB線不平行BC∴中垂線會交於一點
∵△AOD≅△BOD(SAS)
AO=BO
∵△BOE≅△COE(SAS)
CO=BOOA=OB=OC
∵△AOF≅△COF(SSS)
∴∠OFC=∠OFA
∠OFC+∠OFA=180°
∠OFC+∠OFC=180°
2∠OFC=180°
∠OFC=90°
∠OFC=∠OFA=90°∴OFAC
OFAC的中垂線

海倫公式(莊坤霖)

${\displaystyle s={\frac {a+b+c}{2}}}$

△=${\displaystyle {\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}}$
△=${\displaystyle {\frac {1}{2}}}$*a*b*sinr
=${\displaystyle {\frac {1}{2}}ab{\sqrt {1-cos^{2}r}}}$
=${\displaystyle {\frac {1}{2}}ab{\sqrt {1-({\frac {a^{2}+b^{2}-c^{2}}{2ab}})^{2}}}}$
=${\displaystyle {\frac {1}{2}}ab{\sqrt {{\frac {(2ab)^{2}}{(2ab)^{2}}}-{\frac {(a^{2}+b-c)^{2}}{(2ab)^{2}}}}}}$
=${\displaystyle {\frac {1}{2}}ab{\sqrt {\frac {(2ab)^{2}-(a^{2}+b^{2}-c^{2})^{2}}{(2ab)^{2}}}}}$
=${\displaystyle {\frac {1}{2}}ab{\frac {\sqrt {(2ab)-(a^{2}+b^{2}-c^{2})}}{\sqrt {(2ab)^{2}}}}}$
=${\displaystyle {\frac {1}{2}}ab{\frac {\sqrt {(2ab)^{2}-(a^{2}+b^{2}-c^{2})^{2}}}{2ab}}}$
=${\displaystyle {\frac {1}{4}}{\sqrt {(2ab)^{2}-(a^{2}+b^{2}-c^{2})^{2}}}}$
=${\displaystyle {\frac {1}{4}}{\sqrt {[2ab+a^{2}+b^{2}-c^{2}][2ab^{2}-a^{2}+b^{2}-c^{2}]}}}$
=${\displaystyle {\frac {1}{4}}{\sqrt {[(a+b)^{2}-c^{2}][c^{2}-(a^{2}+b^{2}-2ab)]}}}$
=${\displaystyle {\frac {1}{4}}{\sqrt {(a+b+c)(a+b-c)[c-(a-b)][c+(a-b)]}}}$
=${\displaystyle {\sqrt {{\frac {(a+b+c)}{2}}{\frac {(a+b-c)}{2}}{\frac {(c-a+b)}{2}}{\frac {(c+a-b)}{2}}}}}$