In the coordinate plane, point is called lattice point if both of its coordinates are integers. Let A be the point (12, 84). Find the number of right angled triangles ABC in the coordinate plane where B and C are lattice points, having right angle at the vertex A and whose incenter is at the origin (0, 0).
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. We know that if inradius is integer then sides of triangle is also integer.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. Here OA = 60 2.
Hint 3: Proceed with the final algebraic steps to solve the system. Inradius solve for the final value.
Step 1: We know that if inradius is integer then sides of triangle is also integer.
Step 2: Here OA = 60 2.
Step 3: Inradius = 60
Step 4: Hence AB, BC, CA are integer.
Step 5: Let AB = m2 – n2
Step 6: BC = m2 + n2
Step 7: and AC = 2 m2 + n2
Step 8: mn − n
Step 9: Here, = =
Step 10: = ( − ) = 60
Step 11: s 3 (m + n)
Step 12: => n(m – n) = 22.3.5.
Step 13: => Possible values of are 12.
Step 14: Then 12 such triangles are possible but when B and C interchange their position 12
Step 15: more triangles are possible.
Step 16: Then total number of distinct triangles = 24.
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