39. A nice and remarkable problem with Brocard's point.

by Virgil Nicula, May 27, 2010, 2:16 PM

Generalization. Let $ABC$ be a triangle. For a point $P$ consider the points $D\in BC$ , $E\in CA$ , $F\in AB$ so that the quadrilaterals $BFPD$ , $CDPE$ are cyclically.

Denote $m(\angle PDC)=m(\angle PEA)=m(\angle PFB)=\phi$ and $L\in EF\cap BC$ . The sideline $BC$ cut again the circumcircle of $\triangle DEF$ in the point $K$ . Prove that $m\left(\widehat {LP,AK}\right)=\phi$ .

Particular cases.

PC1. Let $\triangle ABC$ and for $P$ let its projections $D\in BC$ , $E\in CA$ , $F\in AB$ on the sidelines of $\triangle ABC$ . Denote

$L\in EF\cap BC$ . The sideline $BC$ cut again the circumcircle of $\triangle DEF$ in $K$ . Prove that $LP\perp AK$ .

PC2. Let $\triangle ABC$ . Denote the midpoint $M$ of $[BC]$ , the projections $E$ , $F$ of orthocenter $H$ to sidelines $AC$ , $AB$ respectively and $L\in BC\cap EF$. Prove that $LH\perp AM$ .

PC3. Let $\triangle ABC$ with incircle $C(I)$ which touches the sides of $\triangle ABC$ in $D\in BC$ , $E\in CA$ , $F\in AB$ . Denote $T\in BC\cap EF$ . Prove that $TI\perp AD$ .

PC4. Let $\triangle ABC$ with exincircle $C(I_a)$ which touches the sides of $\triangle ABC$ in $D'\in BC$ , $E'\in CA$ , $F'\in AB$ . Denote $T'\in BC\cap E'F'$ . Prove that $T'I_a\perp AD'$ .
This post has been edited 4 times. Last edited by Virgil Nicula, Nov 27, 2015, 8:21 AM

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