Tangency Point of Incircle

Question

Is there is a general formula for tangency point of incircle $\triangle{ABC}$ on triangle $\triangle{A}BC$ especially for right triangle

Answer 1

I think what you might be looking for is the Gergonne Point of a triangle. Wikipedia has a good explanation here. The tangency points are given in trilinear coordinates(perpendicular distances to vertices A,B, and C) like so:

$$T_A = 0 : \sec^2(\frac{B}{2}) : \sec^2(\frac{C}{2}) \\ T_B = \sec^2(\frac{A}{2}) : 0 : \sec^2(\frac{C}{2}) \\ T_C = \sec^2(\frac{A}{2}) : \sec^2(\frac{B}{2}): 0$$

Where A,B,C are the relevant angles.