Consider a circle O with a diameter AB, shown here in green. Draw a second circle (red) with diameter AC, such that C is on AB. Draw an isosceles triangle with base CB and third vertex D on circle O. Draw a third circle (X), tangent to the first three figures. Prove that the line from C to the center of circle X is perpendicular to AB.
(In reply to Geometry Theory
Your claim that triangles CFX and CEX are congruent is erroneous, as triangle CFX is not a right triangle. This is because C does not lie on the mutual external tangent of the two circles in question.