315. Indian Team Selection Test 2010 ST1 P1 and extension.

by Virgil Nicula, Aug 31, 2011, 2:00 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

$BFPD$ , $CDPE$ 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 $m\left(\widehat {LP,AK}\right)\in \{\phi ,\pi -\phi\}$ .

Particular cases.

PC1. Let $ABC$ be a triangle. For a point $P$ denote 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 the point $K$ . Prove that $LP\perp AK$ .

PC2.[/size] Let $\triangle ABC$ . Denote the midpoint $M$ of the side $[BC]$ , the projections $E$ , $F$ of the orthocenter $H$

to the sidelines $AC$ , $AB$ respectively and the intersection $L\in BC\cap EF$. Prove that $LH\perp AM$ .

PC3.[/size] Let $ABC$ be a triangle 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.[/size] Let $ABC$ be a triangle 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 $T'I_a\perp AD'$ .

Proof 1 (Luisgeometria). Circles $\odot(BDF)$ , $\odot(CDE)$ , $\odot(AFE)$ concur at their Miquel point $P$ in $\triangle ABC$ . Let $U \equiv AK \cap LP$

and $V$ , $U'$ be the second intersections of $\odot(AFE)$ with $LA$ , $LP$ respectively, then $P$ becomes Miquel point of $\odot(AVE)$ , $\odot(LDV)$ ,

$\odot(CED)$ in $\triangle ALC,$ in other words $P$ , $V$ , $L$ , $D$ are concyclic. In the inversion through pole $L$ with power $LP \cdot LU'$ have $V \mapsto A$ ,

$U' \mapsto P$ and $K \mapsto D$ $\Longrightarrow$ Points $A$ , $U'$ , $K$ are collinear $\Longrightarrow$ $U \equiv U'$ , then $\angle(LP,KA)=\angle(FP,FA)=\phi \ \pmod\pi$ .

Proof 2.
This post has been edited 7 times. Last edited by Virgil Nicula, Nov 20, 2015, 8:18 AM

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