yangqianqiao
幼苗
共回答了14个问题采纳率:100% 举报
在凸四边形内取一点P,使得∠CDP = ∠BDA,∠DCP = ∠DBA.连结PA,PC,PD.
则ΔADB ∽ ΔPDC,∴AB / PC = BD / CD,∴AB * CD = BD * PC
也有AD / DP = BD / CD,
又∵∠ADP = ∠BDC
∴ΔADP ∽ ΔBDC,∴AD / BD = AP / BC,∴AD * BC = BD * AP
∴AB * CD + AD * BC = BD * (AP + PC) ≥ BD * AC.
等号成立当且仅当P在AC上,即∠ACD = ∠PCD = ∠ABD,也就是A、B、C、D四点共圆.
1年前
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