# 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.

The abstract of this article will be published in the August 2012 issue of "Intellectual Archive Bulletin", ISSN 1929-1329.