a, b, and c are sides of a triangle, with a, b, and c are integers. how many triangles which has the perimeter same with its area ?

Question

anonymous · Answer

by using trials i have 2 triangles : (6,8,10) and (5,12,13)
but i dont know if any else ....

anonymous · Answer

@mathmath333 , it is more fun :)

ganeshie8 · Answer

You want to solve  
$$a+b+c = \dfrac{ab\sin C}{2} $$

ganeshie8 · Answer

since the only positive integer values that sin(C) can take are $\{\frac{1}{2}, 1\}$ we get two diophantine equations whose solutions include all the triangles we seek : 

$$a+b+c = \dfrac{ab}{2}\~\a+b+c = \dfrac{ab}{4}$$

ganeshie8 · Answer

looks like there are exactly 5 solutions http://en.wikipedia.org/wiki/Equable_shape

anonymous · Answer

thanks, @ganeshie8