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AB=AC iff CD=BC/2, perpendicular on tangent (Greece Junior 2015)

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July 15, 2019
geometryisoscelestangent

Problem Statement

Let ABCABC be an acute triangle with AB≤ACAB\le AC and let c(O,R)c(O,R) be it's circumscribed circle (with center OO and radius RR). The perpendicular from vertex AA on the tangent of the circle passing through point CC, intersect it at point DD. a) If the triangle ABCABC is isosceles with AB=ACAB=AC, prove that CD=BC/2CD=BC/2. b) If CD=BC/2CD=BC/2, prove that the triangle ABCABC is isosceles.
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