△ABC is a traingle, where O is the incentre and OD is perpendicular to AB. By definition, <OAC = < OAD = θ and < OBC = < OBD = α. Given OD = 3 and BD = 4. The problem is to find the area of △ABC.
My approach : We know the area of a triangle is inradius (r) multiplied with the semicentre (s). So, need to find s. Better to get hold of the sides. Construct the necessary lines to get the following diagram.
We already have r = 3. Now, s would be s = {(a+b)+(a+4)+(b+4)}/2 = a+b+4.
Now, tan(α) = 3/4. Also, a = 3 tan(θ) and b = 3 tan(π/2-α-θ) = 3 cot(α+θ) = 3/tan(α+θ).
Then, s = a+b+4 = 3 tan(θ) + 3/tan(arctan(3/4)+θ)
No progress furthermore. Can anybody suggest any lead? Or maybe some other way to solve the problem.
The unique triangle can not be defined by given data, and the area can be made infinitely big: