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a1267ab   55
N 28 minutes ago by YaoAOPS
Source: IMO Shortlist 2018 G4
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.

Proposed by Mongolia
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a1267ab
Jul 17, 2019
YaoAOPS
28 minutes ago
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Distance between any two points is irrational
orl   23
N 31 minutes ago by ray66
Source: IMO 1987, Day 2, Problem 5
Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
23 replies
orl
Nov 11, 2005
ray66
31 minutes ago
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arbitrary point on Euler line
sarjinius   1
N Apr 22, 2021 by SerdarBozdag
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Suppose that $AB \neq AC$. Let $M$ be an arbitrary point on line segment $OH$. Let $P$ and $Q$ be points on $AB$ and $AC$ respectively such that $MA = MP = MQ$. Prove that the circumcenter of $\triangle MPQ$ lies on the perpendicular bisector of $BC$.
1 reply
sarjinius
Apr 14, 2021
SerdarBozdag
Apr 22, 2021
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arbitrary point on Euler line
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geometry
Orthocenter and Circumcenter
circumcircle
perpendicular bisector
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sarjinius
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sarjinius
#1 h  Apr 14, 2021, 5:57 AM
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Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Suppose that $AB \neq AC$. Let $M$ be an arbitrary point on line segment $OH$. Let $P$ and $Q$ be points on $AB$ and $AC$ respectively such that $MA = MP = MQ$. Prove that the circumcenter of $\triangle MPQ$ lies on the perpendicular bisector of $BC$.
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SerdarBozdag
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SerdarBozdag
#2 h  Apr 22, 2021, 6:08 AM • 1 Y
Y by Mango247
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