向量点乘
注意M在AI中垂线上,IM·(IE+IF)=IE²
因此 E'E* · (2MI+IE) = 2E'F·MI + E'E·IE = 0
取B的旁心J,则 JE=2MI+IE
在AC上取点K使得BR⊥E'E*,即BK//JE
设BK=λJE,其中λ=(B到AC距离)/(B旁切圆半径)=(2RsinAsinC)/[4Rsin(B/2)cos(A/2)cos(C/2)] = 2sin(A/2)sin(C/2)/sin(B/2)
注意内切圆和外接圆内位似,(E',M),(F',N)是对应点,T为位似中心,
TV·BC = TV·BK = TE'·BK = r/(r+R) * ME'·BK
因此
(R+r)/r * TV·BC = ME'·BK = (MI-IE)·λ(2MI+IE) = λ(2MI²-MI·IE-IE²)
=λ(2MI²-ME·IE)
=λ(2MA²-MA·IE)
=λ(2|MA|²+|MA|*r*sin(B/2))
=λ(8R²sin²(B/2)+2Rrsin²(B/2))
=2sin(A/2)sin(B/2)sin(C/2)*(8R²+2Rr)
同理 (R+r)/r * TW·BC 等于这个
注意M在AI中垂线上,IM·(IE+IF)=IE²
因此 E'E* · (2MI+IE) = 2E'F·MI + E'E·IE = 0
取B的旁心J,则 JE=2MI+IE
在AC上取点K使得BR⊥E'E*,即BK//JE
设BK=λJE,其中λ=(B到AC距离)/(B旁切圆半径)=(2RsinAsinC)/[4Rsin(B/2)cos(A/2)cos(C/2)] = 2sin(A/2)sin(C/2)/sin(B/2)
注意内切圆和外接圆内位似,(E',M),(F',N)是对应点,T为位似中心,
TV·BC = TV·BK = TE'·BK = r/(r+R) * ME'·BK
因此
(R+r)/r * TV·BC = ME'·BK = (MI-IE)·λ(2MI+IE) = λ(2MI²-MI·IE-IE²)
=λ(2MI²-ME·IE)
=λ(2MA²-MA·IE)
=λ(2|MA|²+|MA|*r*sin(B/2))
=λ(8R²sin²(B/2)+2Rrsin²(B/2))
=2sin(A/2)sin(B/2)sin(C/2)*(8R²+2Rr)
同理 (R+r)/r * TW·BC 等于这个