The height you have derived is for the triangle seen with the 14 side as its base. That tells us the area of that triangle: 12*14/2=84. Apply the same trick to the 13 edge, and you get another 84, but with an overlap of 5 on the 25 base. That's a similar triangle only scaled down by a third, so volume 84/9. And then there is another similar triangle hanging upside down with base 10, area 4*84/9. Sum it up: 84(2-1/3+4/9)=5*28=140.

I don't see how this "collapses into a perfect square". It would help, I suppose if I were a member and could read the entire piece!