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

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=（h2)^2…②,把(*)式代入②得sina1·sina3=(sina2)^2,∴(-1/2)[cos(a1+a3)-cos(a1-a3)]=(1-cos2a2)/2,∴ cos(a1-a3)=1,∵ a1,a2,a3都是锐角,∴ a1-a3=0,即a1=a3,把它代入①得a1=a2=a3