If p1,p2,p3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then cos⁡Ap1+cos⁡Bp2+cos⁡Cp3 is equal to

We have,cos⁡Ap1+cos⁡Bp2+cos⁡Cp3=12Δ(acos⁡A+bcos⁡B+icos⁡C)=RΔ(sin⁡Acos⁡A+sin⁡Bcos⁡B+sin⁡Ccos⁡C)=R2Δ(sin⁡2A+sin⁡2B+sin⁡2C)=R4sin⁡Asin⁡Bsin⁡C2Δ=2Rsin⁡Asin⁡Bsin⁡CΔ=2RΔ×2Δbc×2Δca×2Δab=16RΔ2a2b2c2=16RΔ2(4RΔ)2=1R