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Circles tangent iff angle BAD is the same as angle CAD

Source: Romanian TST 1997

May 27, 2009
geometrycircumcircleincentergeometric transformationhomothetyratiopower of a point

Problem Statement

Let ABCABC be a triangle, DD be a point on side BCBC, and let O\mathcal{O} be the circumcircle of triangle ABCABC. Show that the circles tangent to O,AD,BD\mathcal{O},AD,BD and to O,AD,DC\mathcal{O},AD,DC are tangent to each other if and only if ∠BAD=∠CAD\angle BAD=\angle CAD.
Dan Branzei
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