设PA = a,PB = b,PC = c,
则:AB^2 = a^2+b^2,BC^2 = b^2 +c^2,AC^2 =a^2 +c^2.
由余弦定理:cos角ABC = [b^2]/[根号(a^2+b^2)(b^2+c^2)]
sin角ABC =根号{ [(a^2+b^2)(b^2+c^2)]-b^4]}/根号[(a^2+b^2)(b^2+c^2)]
三角形ABC的面积为S=(1/2)AB*BC*sin角ABC
=(1/2)根号 [(a^2)(b^2)+(b^2)(c^2)+(c^2)(a^2)]
三角形PAB,PBC,PAC的面积分别为:s1,s2,s3 .
分别有s1= (1/2)a*b,s2=(1/2)bc,s3= (1/2)ac.
由投影定理:s1 = S*cosα,s2= S*cosβ s3 = S*cosγ
即:cosα=(s1)/S,cosβ = (s2)/S cosγ= (s3)/S.
则有:(cosα)^2+(cosβ)^2+(cosγ)^2=[(s1)^2 +(s2)^2 +(s3)^2]/[S^2]
=(1/4)[(a^2)(b^2)+(b^2)(c^2)+(c^2)(a^2)]/{(1/4)[(a^2)(b^2)+(b^2)(c^2)+(c^2)(a^2)]}=1.
即有::(cosα)^2+(cosβ)^2+(cosγ)^2 = 1.