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IMO 2014 Problem 4
ipaper   169
N 37 minutes ago by YaoAOPS
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
169 replies
ipaper
Jul 9, 2014
YaoAOPS
37 minutes ago
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Tangents forms triangle with two times less area
NO_SQUARES   1
N an hour ago by Luis González
Source: Kvant 2025 no. 2 M2831
Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
1 reply
NO_SQUARES
Today at 9:08 AM
Luis González
an hour ago
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Incircle touch points are on circle
zuss77   3
N Jul 16, 2018 by jayme
Source: Kyiv Geometry O.
$\triangle ABC$. $\angle A=90^{\circ}$.
Incircle touches $AB$ and $AC$ at $D, E$.
Middle line parallel to $BC$ cuts $\odot(ABC)$ at $P, T$.
Prove that $D, E, P, T$ are lay on circle.

3 replies
zuss77
Jul 15, 2018
jayme
Jul 16, 2018
J
Incircle touch points are on circle
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Reveal topic tags
geometry
incircle
circumcircle
midline
right triangle
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Source: Kyiv Geometry O.
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zuss77
520 posts
zuss77
#1 h  Jul 15, 2018, 9:13 AM • 2 Y
Y by Adventure10, Mango247
$\triangle ABC$. $\angle A=90^{\circ}$.
Incircle touches $AB$ and $AC$ at $D, E$.
Middle line parallel to $BC$ cuts $\odot(ABC)$ at $P, T$.
Prove that $D, E, P, T$ are lay on circle.
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buratinogigle
2344 posts
buratinogigle
#2 h  Jul 15, 2018, 2:39 PM • 2 Y
Y by Adventure10, Mango247
See https://ijgeometry.com/product/tran-quang-hung-feuerbachs-theorem-right-triangle-extension/
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zuss77
520 posts
zuss77
#3 h  Jul 16, 2018, 7:05 AM • 2 Y
Y by Adventure10, Mango247
Another solution:

Let $Q=DE \cap PT$.
Let $R=AQ \cap BC$.
Let $AQ$ cut $\odot(ABC)$ second time at $K$.

1) $RI \perp AI$ >>

2) $IK \perp AR$ >>

3) $\odot (DEPT)$ >>
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jayme
9782 posts
jayme
#4 h  Jul 16, 2018, 12:16 PM • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,

see : http://jl.ayme.pagesperso-orange.fr/Docs/Quelques%20theoremes%20oublies.pdf

Sincerely
Jean-Louis
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