Mirror on circle line

Question

Let a circle $K$ with center $M(x_m, y_m)$ and radius $r$ be given. Furthermore, two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ outside the circle are given. We are looking for the point $P_3(x_3, y_3)$ on the circumference of the circle, so that the angle between $P_1, P_3$ and the tangent is identical to the angle between $P_2, P_3$ and the tangent. In other words, the tangent is supposed to be a kind of "mirror". You could also rephrase it to say that the angles $\angle P_1P_3M$ and $\angle MP_3P_2$ should be equal. I can derive a formula via the definition of the cosine with the vectors, but I can't get it solved in such a way that I arrive at $x_3$ and $y_3$ via $x_1, y_1, x_2, y_2, x_m, y_m,$ and $r$.