202. Concurrent lines in a right triangle.

by Virgil Nicula, Dec 31, 2010, 8:27 AM

Proposed problem. Let $ABC$ be an $A$-right triangle. For a point $D \in (AC)$ denote the reflection $E$ of $A$

in the line $BD$ and the point $F\in CE$ for which $DF\perp BC$ . Prove that $AF\cap DE\cap BC\ne\emptyset$ .

Proof 1.

Proof 2.
Quote:
Lemma. Let $ w(O)$ , $ w_o$ be two secant circles in $ \{A,B\}$ so that $ O\in w_o$ . Consider a point $P\in (AB)$ and a line $d$ for which $P\in d$ .

Denote $\{M,N\}=w\cap d$ and $R\in w_o$ so that $ N$ separates $ P$ , $ R$ . Then the division $ \{\ M\ ,\ N\ ;\ P\ ,\ R\ \}$ is harmonically.

Particular case. Let $ ABC$ be a triangle with the orthocenter $ H$ . Denote $ D\in BC\cap AH$ and the intersections $ N$ , $ S$

between the line $ AH$ and the circle with the diameter $ [BC]$ . Then the division $ \{\ A\ ,\ H\ ,\ N\ ,\ S\ \}$ is harmonically.

Indeed, $ PO\cdot PR = PA\cdot PB = PM\cdot PN\ \implies\ PO\cdot PR = PM\cdot PN\ \Longleftrightarrow$ the division $ \{\ M\ ,\ N\ ;\ P\ ,\ R\ \}$ is harmonically.


Proof 3.
Quote:
Proposed problem. Let $ w(O)$ , $ w_o$ be two secant circles in $ \{A,B\}$ so that $ O\in w_o$ . Consider a point $P\in (AB)$ and a line $d$ for which $P\in d$ . Denote

$\{M,N\}=w\cap d$ and $R\in w_o$ so that $ N$ separates $ P$ , $ R$ . For $ C\in AB$ (line !) denote $ \left\|\begin{array}{c} X\in MC\cap RA \\ \ Y\in NC\cap RA\end{array}\right\|$ .Then $ MY\cap NX\cap CP\ne\emptyset$ .

Proof.
This post has been edited 19 times. Last edited by Virgil Nicula, Nov 22, 2015, 5:12 PM

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