Question 91719
The equilateral triangle with side PQ is referred Tri_R (opposite to vertex R).
The equilateral triangle with side QR is referred Tri_P (opposite to vertex P).
The equilateral triangle with side RP is referred Tri_Q (opposite to vertex Q).


From Pythagoras' theorem in right angled triangle RPQ,
{{{QR^2 = PQ^2 + RP^2}}} ________ (1)


Now, area of an equilateral triangle with side equal to {{{a}}} is {{{A = sqrt(3)a^2/4}}}.
So, area of Tri_P is {{{A_P = sqrt(3)QR^2/4}}} _____ (2)
So, area of Tri_Q is {{{A_Q = sqrt(3)RP^2/4}}} _____ (3)
So, area of Tri_R is {{{A_R = sqrt(3)PQ^2/4}}} _____ (4)


So, {{{A_Q + A_R = sqrt(3)RP^2/4 + sqrt(3)PQ^2/4}}} [from (3) and (4)]
or {{{A_Q + A_R = sqrt(3)(RP^2 + PQ^2)/4}}}
or {{{A_Q + A_R = sqrt(3)QR^2/4}}} [from (1)]
or {{{A_Q + A_R = A_P}}} [from (2)]