Question

If x, y, z are respectively the distance of the vertices of the $\Delta$ ABC from it's orthocentre , prove that
x + y + z = 2(R + r)

Answer 1

$x+y+z=2R(cosA+cosB+cosC)$
= 2R $[1+4sin\frac{1}{2}Asin\frac{1}{2}Bsin\frac{1}{2}C]$
=$2R+2r=2(R+r).$