PI/BI=△F1IP/△F1IB(同高不同底)
=(F1P*r/2)/(F1B*r/2)=F1P/F1B
=F1P/(F1F2-F2B) .(1)
r为△F1F2P的内切圆半径
同理
PI/BI=△F2IP/△F2IB
=(F2P*r/2)/(F2B*r/2)=F2P/F2B .(2)
由(1),(2)式得
F1P/(F1F2-F2B)=F2P/F2B
F1P*F2B=F2P*F1F2-F2P*F2B.
(F1P+F2P)*F2B=F2P*F1F2 ...(3)
而椭圆
X^2/a^2+Y^2/b^2=1
其离心率
e=c/a ,
而 a^2=b^2+c^2
焦距 F1F2=2c,
而 F1P+F2P=2a
代入(3)
(F1P+F2P)*F2B=F2P*F1F2
2a*F2B=F2P*2c
F2B/F2P=c/a=BI/PI=e