Find the midpoint of a triangle with three endpoint given

Hello,

I try to figure out a formula to find the mid point of a triangle.

Given are the 3 endpoints A(x,y), B(x,y) and C(x,y).

I can find the side length a, b,c of the triangle by using the distance formula.

$\displaystyle d = \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2} $

Then I could find the angles with the Law of Cosine.

$\displaystyle A = \cos^{-1}(\frac{b^2+c^2-a^2}{2bc})$

$\displaystyle B = \cos^{-1}(\frac{a^2+c^2-b^2}{2ac})$

$\displaystyle C = \cos^{-1}(\frac{a^2+b^2-c^2}{2ab})$

The intersection, m(x,y), of the angle bisectors is the center of the incircle.

But how can I find this intersection with the information I have?

Thanks for any help!