Adrian Mitrea. The Six-point Circle Theorem

Natural Sciences / Mathematics / Geometry

Submitted on: Aug 18, 2012, 20:10:57

Description: Given $Delta ABC$ and angles $alpha,beta,gammain(0,pi)$ with $alpha+beta+gamma=pi$, we study the properties of the triangle $DEF$ which satisfies: (i) $Din BC$, $Ein AC$, $Fin AB$, (ii) $aangle D=alpha$, $aangle E=beta$, $aangle F=gamma$, (iii) $Delta DEF$ has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer $Delta DEF$, exists, is unique and is a pedal triangle, corresponding to a certain pedal point $P$. Permuting the roles played by the angles $alpha,beta,gamma$ in (ii), yields a total of six such area-minimizing triangles, which are pedal relative to six pedal points, say, $P_1....,P_6$. The main result of the paper is the fact that there exists a circle which contains all six points.

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