+1306 A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?
+129913 I think there might be a slight mistake above
I think we want A = 3L
So
mn ( m^2 - n^2 ) = 3 [ 2m (m + n) ]
mn ( m + n) ( m - n) = 6m ( m + n) since ( m + n) > 0 we have that
mn ( m - n) = 6m
n ( m - n) = 6
(m - n) = 6 / n
Possibilities (with a little help from WolframAlpha )
m n 2mn m^2 - n^2 m^2 + n^2 Perimeter Area
5 2 20 21 29 70 210 Yes
5 3 30 16 34 80 240 Yes
7 1 14 48 50 112 336 Yes
7 6 84 13 85 182 546 Yes
Added areas = 1332
