1990 OIM Problems/Problem 4

Problem

Let: $C_1$ be a circle, $AB$ be one of its diameters, $t$ be its tangent at $B$ and $M$ be a point on $C_1$ other than $A$. A circle $C_2$ is constructed tangent to $C_1$ at $M$ and to the line $t$.

a. Determine the point $P$ of tangency of $t$ and $C_2$, and find the locus of the centers of the circles by varying $M$.

b. Prove that there is a circle orthogonal to all the circles $C_2$.

NOTE: Two circles are orthogonal if they intersect and the respective tangents at the points of intersection are perpendicular.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

https://www.oma.org.ar/enunciados/ibe5.htm