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SOAB)+S(OAC)=2S(OBC)

Source: Iran 2005

August 27, 2005
geometrycircumcircleEulerpower of a pointradical axisgeometry proposed

Problem Statement

Suppose OO is circumcenter of triangle ABCABC. Suppose S(OAB)+S(OAC)2=S(OBC)\frac{S(OAB)+S(OAC)}2=S(OBC). Prove that the distance of OO (circumcenter) from the radical axis of the circumcircle and the 9-point circle is a29R2−(a2+b2+c2)\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}
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