OPM 2009 Problem 5

Circumferences $C_1$ and $C_2$ have different radios and are externally tangent on point $T$ . Consider points $A$ on $C_1$ and $B$ on $C_2$ , both different from $T$ , such that $\angle BTA=90^{\circ}$ . What is the locus of the midpoints of line segments $AB$ constructed that way?