If AB is a common tangent to two circles, prove that the circle on AB as a diameter cuts each of the circles orthogonally.
Source: Challenge and Thrills in Pre College Mathematics.
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Proving Orthognality at points A and B is trivial as it follows from the definition of a tangent (and the angle that it forms with the radius of the circle drawn through the point of contact).
What remains to be proven is that the other two points of intersection A' and B' of the circle with Diameter AB with the original two circles are such that OA' and OB' are tangents where O is the centre of the third circle.
Let P be the centre of one of the circles whose tangents are drawn OA and OA' It is easy to prove that triangle OAP and OA'P are congruent by S-S-S test. Hence, angle OAP=angle OA'P=90 degrees. Thus, OA' is also a tangent, which completes the proof.