Inspired by pennypc123456789
by sqing, Apr 7, 2025, 9:28 AM
Let 
Prove that
Prove that
isogonal geometry
by Tuguldur, Apr 7, 2025, 4:27 AM
Let 
and 
be isogonal conjugates with respect to 
. Let 
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](//latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
, 
and 
lie on the line 
.
and 
be isogonal conjugates with respect to 
. Let 
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
, 
and 
lie on the line 
.Geometry
by youochange, Apr 6, 2025, 11:27 AM
m:}
Let be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point . Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point , and intersects the circumcircle of again at point 
.
Prove that the point lies on the circumcircle of 
.
Let
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point . Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point , and intersects the circumcircle of again at point 
.Prove that the point
lies on the circumcircle of 
.This post has been edited 1 time. Last edited by youochange, Yesterday at 11:28 AM
Reason: Y
Reason: Y
Incenter and concurrency
by jenishmalla, Mar 15, 2025, 2:40 PM
Let the incircle of touch sides , 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.
(Kritesh Dhakal, Nepal)
touch sides , 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.(Kritesh Dhakal, Nepal)
This post has been edited 2 times. Last edited by jenishmalla, Mar 15, 2025, 3:00 PM
Reason: formatting
Reason: formatting
All Russian Olympiad Day 1 P4
by Davrbek, Apr 28, 2018, 10:36 AM
On the sides and of the triangle 
, the points and are chosen, respectively, so that 
. Segments 
and 
intersect at point 
. Point 
is symmetric to point 
relative to line . The segment 
intersects the circumcircle 
of the triangle 
at the point 
. Prove that circumcircle of 
is tangent to the circle .
and of the triangle 
, the points and are chosen, respectively, so that 
. Segments 
and 
intersect at point 
. Point 
is symmetric to point 
relative to line . The segment 
intersects the circumcircle 
of the triangle 
at the point 
. Prove that circumcircle of 
is tangent to the circle .This post has been edited 3 times. Last edited by djmathman, Dec 18, 2018, 7:56 PM
Reason: formatting + wording
Reason: formatting + wording
Find < BAC given MB = OI
by math163, Nov 11, 2017, 2:14 PM
Let be a triangle in which 
. Let 
and be the incentre and circumcentre of , respectively. Let be the midpoint of the arc of the circumcircle of , which does not contain the point . Determine 
given that 
.
be a triangle in which 
. Let 
and be the incentre and circumcentre of , respectively. Let be the midpoint of the arc of the circumcircle of , which does not contain the point . Determine 
given that 
.This post has been edited 6 times. Last edited by math163, Aug 15, 2018, 6:36 PM
Geometry
by IstekOlympiadTeam, Dec 12, 2015, 4:33 PM
An acute-angled 
is inscribed into a circle 
. Let be the centroid of , and let 
be an altitude of this triangle. A ray 
meets at . Prove that the circumcircle of the triangle 
is tangent to . (A.I. Golovanov , A.Yakubov)
is inscribed into a circle 
. Let be the centroid of , and let 
be an altitude of this triangle. A ray 
meets at . Prove that the circumcircle of the triangle 
is tangent to . (A.I. Golovanov , A.Yakubov)Not homogenous
by Arne, Mar 23, 2004, 7:21 PM
Prove that the inequality ![\[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\]](//latex.artofproblemsolving.com/e/9/a/e9a57f0700d37fbb28f7d219cc7ac7e52b43acd9.png)
holds for all positive reals 
, 
, 
.
![\[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\]](http://latex.artofproblemsolving.com/e/9/a/e9a57f0700d37fbb28f7d219cc7ac7e52b43acd9.png)
holds for all positive reals 
, 
, 
.This post has been edited 3 times. Last edited by djmathman, Sep 15, 2015, 12:52 PM
Reason: formatting
Reason: formatting
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
April 2021
February 2021
March 2020
February 2018
January 2018
November 2017
September 2017
July 2017
June 2017
May 2017
April 2017
March 2017
February 2017
December 2016
April 2016