Pythagorean Fun!

[Denis]

Pythagorean triangle ABC with area 8214
has square DEFG with sides = 60 inscribed in it.
Side DE of the square lies on the hypotenuse.
Find the triangle's side lengths.

B

         E


                     D

F



C         G                               A

 

[Otis]

I started with triangle CGF isosceles (which gives 8100 for area of triangle ABC) and then rotated the square, using angle CGF as a parameter. With symmetry identities, the adjustments to AC and BC were easy enough, but then I got lazy and used technology to solve my equation.

My answer: 111, 148, 185

 

[Denis]

Yepper!
a = BC, b=AC, c = AB, d = square side
Formula:
d = abc / (a^2 + b^2 + ab)
111*148*185 / (111^2 + 148^2 + 111*148) = 60

However, wrote looper program to solve...