>>419
内心は
I = {sin(A)*A + sin(B)*B + sin(C)*C}/{sin(A)+sin(B)+sin(C)},

sin(A)/{sin(A)+sin(B)+sin(C)} = 1/{2cos(B)} + 1/{2cos(C)} - σ{tan(B)+tan(C)}/2,
等を入れて
I = {1/cos(C) - σtan(C)}*(A+B)/2 + ･･･
= {1-σsin(C)}*M + {1-σsin(A)}*N + {1-σsin(B)}*(-x),

ここに、(A+B)/2=cos(C)*M、(B+C)/2=cos(A)*N、(C+A)/2=cos(B)*(-x),
σ = {tan(A/2)+tan(B/2)+tan(C/2)+tan(A/2)tan(B/2)tan(C/2)}/2.

一方、外心は
O = {sin(2A)*A + sin(2B)*B +sin(2C)*C}/{sin(2A)+sin(2B)+sin(2C)}
= {tan(C)*(A+B)/2 + tan(A)*(B+C)/2 + tan(B)*(C+A)/2}/{tan(A)+tan(B)+tan(C)}
= {sin(C)*M + sin(A)*N +sin(B)*(-x)}/{tan(A)+tan(B)+tan(C)}
= 0,
∴　σの項は消えて
I = M + N + (-x),
3傍心も同様。

ぬるぽ