设三棱锥A-BCD棱长为x,两棱之间夹角为θ,一棱与其对面夹角为α,即题述中的a.
以V（A-BCD）表示三棱锥A-BCD的体积,V（A-MNF）表示三棱锥A-MNF的体积,则有
[V（A-MNF）]/[V（A-BCD）]=(AM·AN·AF)/(x^3)
而V（A-BCD）=(1/3)[(1/2)(x^2)(sinθ)](xsinα)———
{中括号内为A-BCD侧面积,最后的小括号内为对应的高}
由以上两式得V（A-MNF)=(1/6)(AM·AN·AF)(sinα)(sinθ)
又有V（A-MNF)=V（E-AMN)+V（E-ANF)+V（E-AMF)=(1/3)[(1/2)(AM·AN)(sinθ)]*d+(1/3)[(1/2)(AN·AF)(sinθ)]*d+(1/3)[(1/2)(AF·AM)(sinθ)]*d——{中括号内依次为△AMN、△ANF、△AMF面积}
由上(1/3)[(1/2)(AM·AN)(sinθ)]*d+(1/3)[(1/2)(AN·AF)(sinθ)]*d+(1/3)[(1/2)(AF·AM)(sinθ)]*d=(1/6)(AM·AN·AF)(sinα)(sinθ)
整理出1/AM+1/AN+1/AF=sinα/d