TITLE: As shown in the figure: Prove that $a^2+b^2=c^2$
QUESTION [17 upvotes]: Geometry: Buildings in the triangle

Other triangles with the same property:
$1.$  12  18   6  12  30 102
$2.$  15  30  15  15  15  90
$3.$  24  30  54  24   6  42
$4.$  30  10  40  30  20  50 (proposed problem in sense of clockwise)
$5.$  36  12   6  12  18  96
$6.$  36  18   6  36   6  78
$7.$  42   6  36  42  12  42
$8.$  60   6  57  30   3  24
$9.$  60  24  12  12   6  66        
Using matlab we can find all triangles (integer solutions) with this property sums of squares:

REPLY [6 votes]: I thought that this problem should be solvable without using trigonometry. Here's a hint for a geometric solution:

Draw the segment $DG$ and let $B'$ be the intersection of the line through $AE$ and the perpendicular line to $DG$. I claim that the triangle $DB'G$ has sides of length $a,b,c$, so $c^2 = a^2 + b^2$ by the Pythagorean theorem.
Since geometry is hard to communicate, I think it is better to let you figure this out on your own.