M(x,y) is on the circle iff
M is on the line iff
M is on the circle and on the line iff (x,y) satisfy the system
To solve you just need to substitute y in the first equation using the second equation. You will get a quadratic equation (unknown x) which can have 0 (no intersection point), 1 (1 intersection point = the line and the circle are tangent) or 2 solutions depending by the value of the discriminant.
As soon as you have found the values for x, you substitute in the second equation to get the values for y.
A(13,13), B(9,3), and C(-1,-1).
M being the midpoint of (BC) : and
(AM) passes through A(13,13) and M(4,1). You should be able to find its equation now.
As soon as you have the equation of (AM) and (BN) you can solve the system to get the coordinates of P.