G
Topic
First Poster
Last Poster
orthocenter on sus circle
DVDTSB 1
N
by Double07
Source: Romania TST 2025 Day 2 P1
Let 
be an acute triangle with 
, and let 
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
. The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point 
. Let 
be a point on the ray 
, starting at , such that 
.
Show that the orthocenter of triangle lies on the circle with diameter 
.
Proposed by Radu Lecoiu
be an acute triangle with 
, and let 
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
. The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point 
. Let 
be a point on the ray 
, starting at , such that 
.Show that the orthocenter of triangle
lies on the circle with diameter 
.Proposed by Radu Lecoiu
1 reply
+1 w
Incircle triangles inequality
MathMystic33 0
Source: 2025 Macedonian Team Selection Test P5
Let 
be a triangle with side‐lengths 
, incenter 
, and circumradius 
. Denote by 
the area of , and let 
be the areas of triangles 
, 
, and 
, respectively. Prove that
![\[
\frac{abc}{12R}
\;\le\;
\frac{P_1^2 + P_2^2 + P_3^2}{P}
\;\le\;
\frac{3R^3}{4\sqrt[3]{abc}}.
\]](//latex.artofproblemsolving.com/1/3/9/139a3e15f08afb78ca0b8301f24d30db8f202207.png)
be a triangle with side‐lengths 
, incenter 
, and circumradius 
. Denote by 
the area of , and let 
be the areas of triangles 
, 
, and 
, respectively. Prove that![\[
\frac{abc}{12R}
\;\le\;
\frac{P_1^2 + P_2^2 + P_3^2}{P}
\;\le\;
\frac{3R^3}{4\sqrt[3]{abc}}.
\]](http://latex.artofproblemsolving.com/1/3/9/139a3e15f08afb78ca0b8301f24d30db8f202207.png)
0 replies
Collinearity of intersection points in a triangle
MathMystic33 0
Source: 2025 Macedonian Team Selection Test P1
On the sides of the triangle 
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let
![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](//latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
0 replies
Brazilian Locus
kraDracsO 15
N
by Ilikeminecraft
Source: IberoAmerican, Day 2, P4
Let 
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.
Note: The locus is the set of all points of the plane that satisfies the property.
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.Note: The locus is the set of all points of the plane that satisfies the property.
15 replies
Circumcircle of MUV tangent to two circles at once
MathMystic33 0
Source: Macedonian Mathematical Olympiad 2025 Problem 1
Given is an acute triangle 
with . Let be the midpoint of side , and let 
and 
be points on segments 
and 
, respectively, such that 
. Let 
be the circumcircle of 
, and 
the circumcircle of 
. The common tangent 
to and , which lies closer to point , touches and at points 
and 
, respectively. Let the line 
intersect again at 
, and the line 
intersect again at 
. Prove that the circumcircle of triangle 
is tangent to both and .
with . Let be the midpoint of side , and let 
and 
be points on segments 
and 
, respectively, such that 
. Let 
be the circumcircle of 
, and 
the circumcircle of 
. The common tangent 
to and , which lies closer to point , touches and at points 
and 
, respectively. Let the line 
intersect again at 
, and the line 
intersect again at 
. Prove that the circumcircle of triangle 
is tangent to both and .0 replies
Aime 2005a #15
4everwise 22
N
by Ilikeminecraft
Source: Aime 2005a #15
Triangle has 
. The incircle of the triangle evenly trisects the median 
. If the area of the triangle is 
where 
and 
are integers and is not divisible by the square of a prime, find 
.
has 
. The incircle of the triangle evenly trisects the median 
. If the area of the triangle is 
where 
and 
are integers and is not divisible by the square of a prime, find 
.
22 replies
Nice geometry...
Sadigly 1
N
by aaravdodhia
Source: Azerbaijan Senior NMO 2020
Let be a scalene triangle, and let be its incenter. A point 
is chosen on line , such that the circumcircle of triangle 
intersects 
at 
, and the circumcircle of triangle 
intersects 
at 
. Circumcircle of triangle 
intersects and at 
and 
, respectively. Lines 
and 
intersect at 
, and lines 
and 
intersect at . Prove that 
.
be a scalene triangle, and let be its incenter. A point 
is chosen on line , such that the circumcircle of triangle 
intersects 
at 
, and the circumcircle of triangle 
intersects 
at 
. Circumcircle of triangle 
intersects and at 
and 
, respectively. Lines 
and 
intersect at 
, and lines 
and 
intersect at . Prove that 
.
1 reply
AM=CN in Russia
mathuz 25
N
by Ilikeminecraft
Source: AllRussian-2014, Grade 11, day1, P4
Given a triangle with 
, 
is circumcircle. Let , are lie on the sides , respectively, such that 
. 
and is incenter of the triangle 
, is K-excenter of the triangle 
(opposite to 
and tangents to 
). If is midpoint of the arc of then prove that 
.
M. Kungodjin
with 
, 
is circumcircle. Let , are lie on the sides , respectively, such that 
. 
and is incenter of the triangle 
, is K-excenter of the triangle 
(opposite to 
and tangents to 
). If is midpoint of the arc of then prove that 
.M. Kungodjin
25 replies
Simson lines on OH circle
DVDTSB 2
N
by SomeonesPenguin
Source: Romania TST 2025 Day 2 P4
Let and 
be two triangles inscribed in the same circle, centered at , and sharing the same orthocenter 
. The Simson lines of the points 
with respect to triangle form a non-degenerate triangle 
.
Prove that the orthocenter of lies on the circle with diameter 
.
Note. Assume that the points
lie in this order on the circle and form a convex, non-degenerate hexagon.
Proposed by Andrei Chiriță
and 
be two triangles inscribed in the same circle, centered at , and sharing the same orthocenter 
. The Simson lines of the points 
with respect to triangle form a non-degenerate triangle 
.Prove that the orthocenter of
lies on the circle with diameter 
.Note. Assume that the points
lie in this order on the circle and form a convex, non-degenerate hexagon.Proposed by Andrei Chiriță
2 replies
