Question

平面几何高手速来!P是△ABC内一点,AP、BP、CP分别交对边于D、E、F

Answer 1

由面积比等于边长比

(S1+ S5 + S6) / S1 = (S2 + S3 + S4) / S2 推出

S2S5 + S2S6 = S1S4 + S1S3

同理

S1S4 + S2S4 = S3S6 + S3S5

S4S6 + S3S6 = S2S5 + S1S5

三式左右相加消去相同项得

S1S3 + S3S5 + S1S5 = S2S4 + S4S6 + S2S6 等式1

又由西瓦定理知(S1/S2) *( S3/S4) * (S5/S6) = 1

推出 S1S3S5 = S2S4S6 等式2

等式1 除以 等式2 得出

1/S1 + 1/S3 + 1/S5 = 1/S2 + 1/S4 + 1/S6