An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.
Problem
Source: Sharygin 2023 - P23 (Grade-10-11)
Tags: geometry, ellipse, de longchamps point, collinearity, Sharygin Geometry Olympiad, Sharygin 2023
sanyalarnab
04.03.2023 06:26
link to this problem quite weird seeing this problem posted 3 days before the start of the test.
DottedCaculator
04.03.2023 09:31
another relevant thread
LoloChen
04.03.2023 11:45
This is a well-known problem.
kamatadu
04.03.2023 12:19
Well, even worse right here where Luiz Gonzalez SAAR himself posted the proof using Soddy Lines .
starchan
04.03.2023 13:34
I had a different proof, where you find an interesting understanding of the common line through two confocal ellipses. This is probably well known but I'll still post my proof.
Attachments:
Sharygin-Problem-23.pdf (208kb)
thevictor
11.04.2023 19:04
Denote by $l_1,l_2$ the directrix of the eclipses. We have (let $X,Y$ be the two intersections) $\frac{XM_b}{d_{x-l_1}}=e_1,\frac{XM_b}{d_{x-l_2}}=e_2$,so $\frac{d_{x-l_1}}{d_{x-l_2}}=\frac{e_2}{e_1}$,the same as $Y$,so it suffices to verify $H$ for this, which is trivial.