MathDB

2

Part of 2002 Turkey MO (2nd round)

Problems(2)

A is the incenter if the centers of circles aren't collinear

Source: Turkey National Olympiad 2002 - D1 - P2

3/11/2011
Two circles are externally tangent to each other at a point AA and internally tangent to a third circle Γ\Gamma at points BB and C.C. Let DD be the midpoint of the secant of Γ\Gamma which is tangent to the smaller circles at A.A. Show that AA is the incenter of the triangle BCDBCD if the centers of the circles are not collinear.
geometryincentercircumcircleangle bisectorgeometry unsolved
Equal to distance between the incenter and the circumcenter

Source: Turkey National Olympiad 2002 - D2 - P2

3/6/2011
Let ABCABC be a triangle, and points D,ED,E are on BA,CABA,CA respectively such that DB=BC=CEDB=BC=CE. Let O,IO,I be the circumcenter, incenter of △ABC\triangle ABC. Prove that the circumradius of △ADE\triangle ADE is equal to OIOI.
geometryincentercircumcircleratiogeometry proposed