In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. Let A be the point (12,84) . Find the number of right angled triangles A B C in the coordinate plane where B and C are lattice points, having a right angle at the vertex A and whose incenter is at the origin (0,0) .

Step-by-step solution

We know that if inradius is integer then sides of triangle is also integer. Here O A=60 2 . Inradius =60 Hence A B, B C, C A are integer. Let A B=m^2-n^2B C=m^2+n^2 and A C=2 m^2+n^2 Here, r= s = m n (m^2-n^2 ) 3(m+n) =n(m-n)=60 n(m-n)=2^2 .3 .5 . Possible values of n are 12 . Then 12 such triangles are possible but when B and C interchange their position 12 more triangles are possible. Then total number of distinct triangles =24 .