Indonesia National Science Olympiad 2010 - Day 1 Problem 2

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Indonesia National Science Olympiad 2010 - Day 1 Problem 2

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projective geometry

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#1 h Sep 28, 2010, 5:14 AM • 3 Y

Y by brain_idea, Adventure10, Mango247

Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$ . The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$ , respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$ . Show that the line $CO$ , the line which passed through $F$ and perpendicular to $AC$ , and the line which passed through $E$ and parallel with $DO$ are concurrent.

Fajar Yuliawan, Bandung

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#2 h Sep 28, 2010, 4:08 PM • 2 Y

Y by Adventure10, Mango247

Denote $R$ the intersection of $CO$ with the perpendicular line to $AC$ through $F.$ Then it is enough to show that $ER \parallel DO.$ Let the ray $CO$ cut $AB$ and the circumcircle $(O)$ at $P,Q,$ respectively. Since $QA \perp CA,$ it follows that $\triangle CFR$ and $\triangle CAQ$ are homothetic, and their cevians $FO$ and $AP$ are homologous $\Longrightarrow$ $\frac{_{CP}}{^{PQ}}=\frac{_{CO}}{^{OR}} \ (*).$ Because of $\angle ACQ=\angle BCE,$ it follows that $EQ \parallel AB.$ Hence, by Thales theorem $\frac{_{CP}}{^{PQ}}=\frac{_{CD}}{^{DE}},$ together with $(*)$ we obtain $\frac{_{CO}}{^{OR}}=\frac{_{CD}}{^{DE}}$ $\Longrightarrow$ $ER \parallel DO.$

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jayme

#3 h Sep 29, 2010, 5:35 AM • 1 Y

Y by Adventure10

Dear Mathlinkers,
following the notation of Luis,
1. let P the point of intersection of the parallel to B passing through O with CED.
2. According to the Desargues' week theorem applied to the homothetic triangles FRB and AQD (center C), RP // QD.
3. According to the little Pappus theorem applied to the hexagone REQDOPR, RE//DO.
Sincerely
Jean-Louis

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#4 h Nov 13, 2021, 2:57 AM

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My solution in great detail, very beginner-friendly. Enjoy!

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