Problem 9:Notes on Eucledian Geometry,radical axis,problem 1

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by henderson, Feb 24, 2016, 4:58 PM

$\bf\color{red}Problem \ 9 \$ Let $ABC$ be a triangle. Let the points $D,$ $E,$ $F$ be on the perpendicular bisectors of $BC,$ $CA,$ $AB,$ respectively. Show that the lines through $A,$ $B,$ $C$ perpendiculars to $EF,$ $FD,$ $DE,$ respectively are concurrent.
$\bf\color{red}Solution \$ Let the points be $K,$ $L,$ $M,$ respectively $,$ where the perpendiculars from $A,$ $B,$ $C$ meet the lines $EF,$ $FD,$ $ED$ and let the points where $OK,$ $OE,$ $OD$ meet $AB,$ $AC,$ $BC$ be $M,$ $N,$ $P.$
$($ $O$ is the circumcentre. $)$
Using radical axis theorem for the circumcircles of $AKFM,$ $BLFM,$ $BLDP,$ $DPCM,$ $DCNE,$ $ENAK,$ the conclusion follows.

This post has been edited 5 times. Last edited by henderson, Sep 12, 2016, 2:34 PM

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Problem 9:Notes on Eucledian Geometry,radical axis,problem 1

by henderson, Feb 24, 2016, 4:58 PM

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