68. Some nice applications of "pole/polar".

by Virgil Nicula, Jul 24, 2010, 1:31 AM

PP1. Let $ABC$ be a triangle with incircle $w$ . Let $w_{1}$ be the circle through $B$, $C$ and tangent to $w$ . Define analogously

$w_{2}$ , $w_{3}$ . Let $T_{i}$ be the tangency point of $w_{i}$ and $w$ , where $k\in \overline {1,2,3}$ . Prove that $AT_{1}$ , $BT_{2}$ , $CT_{3}$ are concurrently.

Proof. Let the common tangent $\tau_a$ of $\gamma$ and $\gamma_1$ cut $BC$ at $X$ . We define $Y$ , $Z$ on $CA$ , $AB$ as $X$ . Since $\tau_a$ is the radical axis of $\gamma$ ,$\gamma_1$ , it follows that

${XT_1}^2=XB \cdot XC$ $\Longrightarrow$ $X$ has equal power WRT $\gamma$ and circumcircle $\omega$ of $\triangle ABC$ . Likewise, $Y$ , $Z$ have equal power w.r.t. $\gamma$ and $\omega$ . Hence, $X$ , $Y$ , $Z$

are collinear on the radical axis of $\gamma,\omega$ . If $D,E,F$ are the tangency points of $\gamma$ with $BC,CA,AB$ , then $X$ , $Y$ , $Z$ become poles of lines $DT_1$ , $ET_2$ and $FT_3$

w.r.t. $\gamma$ $\Longrightarrow$ $DT_1$ , $ET_2$ and $FT_3$ concur at the pole $U$ of $XYZ$ WRT $\gamma$ . Now, due to Steinbart's theorem we conclude that $AT_1$ , $BT_2$ , $CT_3$ are concurrent.


PP2. Let $ABCD$ be a quadrilateral inscribed in the circle $w=C(O,R)$ . Denote $\left\{\begin{array}{c} P\in AB\cap CD\\\ Q\in AD\cap BC\end{array}\right|$ . Prove that $ \overrightarrow{OP} \cdot \overrightarrow{OQ}= R^2$ .

Proof. Let $E \equiv AC \cap BD$ . Since the pencil $Q(A,B,E,P)$ is harmonically, it follows that $QE$ is the polar of $P$ w.r.t. $w$ $\Longrightarrow$ $QE\perp OP$ at $S$ .

If $\{M,N\}=QE \cap w$ , then $PM$ and $PN$ are tangents to $w$ $\Longrightarrow$ $OM^2=R^2=|OS| \cdot |OP|=|OP| \cdot |OQ| \cdot \cos \angle POQ=\overrightarrow{OP} \cdot \overrightarrow{OQ}$ .
This post has been edited 15 times. Last edited by Virgil Nicula, Nov 23, 2015, 4:06 PM

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