三棱锥P-ABC,顶点到BC,AC,AB的距离为h1.h2.h3.二面角P-BC-1=A1.P-AC-B=A2,P-AB-C=A3.a1,a2,a3都是锐角依次成等差,h1,h2,h3依次成等比.求证

题目

三棱锥P-ABC,顶点到BC,AC,AB的距离为h1.h2.h3.二面角P-BC-1=A1.P-AC-B=A2,P-AB-C=A3.a1,a2,a3都是锐角依次成等差,h1,h2,h3依次成等比.求证a1=a2=a3

答案

设P在面ABC的射影是O,且PO=h,OD⊥BC于D,OE⊥AC于E,OF⊥AB于F,则sina1=h/h1,sina2=h /h2,sina3=h/h3,∴ h1=h/sina1,h2=h/sina2,h3=h/sina3…（*）,又a1,a2,a3成等差数列,h1h2h3成等比数列,∴ a1+a3=2a2…①,h1·h3=（...