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This post has been edited 5 times. Last edited by B1t, Apr 30, 2025, 6:56 AM
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Another right angled triangle
ariopro1387 6
N
by sami1618
Source: Iran Team selection test 2025 - P7
Let 
be a right angled triangle with 
.Let 
be the midpoint of 
, and 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
be a right angled triangle with 
.Let 
be the midpoint of 
, and 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
6 replies
A weird problem
jayme 2
N
by lolsamo
Dear Mathlinkers,
1. ABC a triangle
2. 0 the circumcircle
3. I the incenter
4. 1 a circle passing througn B and C
5. X, Y the second points of intersection of 1 wrt BI, CI
6. 2 the circumcircle of the triangle XYI
7. M, N the symetrics of B, C wrt XY.
Question : if 2 is tangent to 0 then, 2 is tangent to MN.
Sincerely
Jean-Louis
1. ABC a triangle
2. 0 the circumcircle
3. I the incenter
4. 1 a circle passing througn B and C
5. X, Y the second points of intersection of 1 wrt BI, CI
6. 2 the circumcircle of the triangle XYI
7. M, N the symetrics of B, C wrt XY.
Question : if 2 is tangent to 0 then, 2 is tangent to MN.
Sincerely
Jean-Louis
2 replies
Channel name changed
Plane_geometry_youtuber 10
N
by Yiyj
Hi,
Due to the search handle issue in youtube. My channel is renamed to Olympiad Geometry Club. And the new link is as following:
https://www.youtube.com/@OlympiadGeometryClub
Recently I introduced the concept of harmonic bundle. I will move on to the conjugate median soon. In the future, I will discuss more than a thousand theorems on plane geometry and hopefully it can help to the students preparing for the Olympiad competition.
Please share this to the people may need it.
Thank you!
Due to the search handle issue in youtube. My channel is renamed to Olympiad Geometry Club. And the new link is as following:
https://www.youtube.com/@OlympiadGeometryClub
Recently I introduced the concept of harmonic bundle. I will move on to the conjugate median soon. In the future, I will discuss more than a thousand theorems on plane geometry and hopefully it can help to the students preparing for the Olympiad competition.
Please share this to the people may need it.
Thank you!
10 replies
Euler line of incircle touching points /Reposted/
Eagle116 6
N
by pigeon123
Let be a triangle with incentre 
and circumcentre 
. Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
be a triangle with incentre 
and circumcentre 
. Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
6 replies
Parallel lines on a rhombus
buratinogigle 1
N
by Giabach298
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Given the rhombus 
with its incircle 
. Let 
and be the points of tangency of with and respectively. On the edges 
and 
, take points 
and 
such that 
is tangent to at . Suppose is the intersection point of the lines 
and 
. Prove that two lines 
and 
are parallel or coincide.
with its incircle 
. Let 
and be the points of tangency of with and respectively. On the edges 
and 
, take points 
and 
such that 
is tangent to at . Suppose is the intersection point of the lines 
and 
. Prove that two lines 
and 
are parallel or coincide.
1 reply
Orthocenter lies on circumcircle
whatshisbucket 90
N
by bjump
Source: 2017 ELMO #2
Let be a triangle with orthocenter 
and let be the midpoint of 
Suppose that and are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
Proposed by Michael Ren
be a triangle with orthocenter 
and let be the midpoint of 
Suppose that and are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
Proposed by Michael Ren
90 replies
Hagge-like circles, Jerabek hyperbola, Lemoine cubic
kosmonauten3114 0
Source: My own
Let 
be a scalene oblique triangle with circumcenter and orthocenter , and (
, 
) a point in the plane.
Let
, 
be the circumcevian triangles of , , respectively.
Let
be the pedal triangle of with respect to .
Let
be the reflection in 
of 
. Define 
, 
cyclically.
Let
be the reflection in of 
. Define 
, 
cyclically.
Let
, 
be the circumcenters of 
, 
, respectively.
Prove that:
1), , are collinear if and only if lies on the Jerabek hyperbola of .
2), , are collinear if and only if lies on the Lemoine cubic (= 
) of .
be a scalene oblique triangle with circumcenter and orthocenter , and (
, 
) a point in the plane.Let
, 
be the circumcevian triangles of , , respectively.Let
be the pedal triangle of with respect to .Let
be the reflection in 
of 
. Define 
, 
cyclically.Let
be the reflection in of 
. Define 
, 
cyclically.Let
, 
be the circumcenters of 
, 
, respectively.Prove that:
1)
, , are collinear if and only if lies on the Jerabek hyperbola of .2)
, , are collinear if and only if lies on the Lemoine cubic (= 
) of .
0 replies
Incenter perpendiculars and angle congruences
math154 84
N
by zuat.e
Source: ELMO Shortlist 2012, G3
is a triangle with incenter . The foot of the perpendicular from to is 
, and the foot of the perpendicular from to 
is . Prove that 
.Alex Zhu.
84 replies
Tangency of circles with "135 degree" angles
Shayan-TayefehIR 4
N
by Mysteriouxxx
Source: Iran Team selection test 2024 - P12
For a triangle 
with an obtuse angle 
, let 
be feet of altitudes from 
on sides 
respectively. The tangents from to circumcircle of triangle intersect line 
at points 
respectively and we know that 
. Point 
lies on segment 
in such a way that 
and let 
be a point on line such that 
is between 
and 
. Prove that the circle with diameter 
is tangent to circumcircle of triangle .
Proposed by Mehran Talaei
with an obtuse angle 
, let 
be feet of altitudes from 
on sides 
respectively. The tangents from to circumcircle of triangle intersect line 
at points 
respectively and we know that 
. Point 
lies on segment 
in such a way that 
and let 
be a point on line such that 
is between 
and 
. Prove that the circle with diameter 
is tangent to circumcircle of triangle .Proposed by Mehran Talaei
4 replies
Line bisects a segment
buratinogigle 1
N
by cj13609517288
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Let be a triangle with 
. A circle 
is tangent to sides and , and is the midpoint of . Points and lie on sides and , respectively, such that segment is tangent to circle at point . Let and be the orthocenters of triangles 
and 
, respectively. Prove that line 
bisects segment 
.
be a triangle with 
. A circle 
is tangent to sides and , and is the midpoint of . Points and lie on sides and , respectively, such that segment is tangent to circle at point . Let and be the orthocenters of triangles 
and 
, respectively. Prove that line 
bisects segment 
.1 reply
Three collinear points
buratinogigle 1
N
by Giabach298
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Let be a triangle with points and lying on rays and , respectively, such that 
. On the line , choose points and 
such that 
and 
. Let and 
be the feet of the perpendiculars from and to line , respectively. Let 
be the intersection point of lines 
and 
. Prove that the reflection of over line lies on the line connecting 
and the circumcenter of triangle .
be a triangle with points and lying on rays and , respectively, such that 
. On the line , choose points and 
such that 
and 
. Let and 
be the feet of the perpendiculars from and to line , respectively. Let 
be the intersection point of lines 
and 
. Prove that the reflection of over line lies on the line connecting 
and the circumcenter of triangle .1 reply
weird Condition
G
H
G
H
Source: Mongolian TST 2025 P4
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Y by
Let midpoint of arc 
on 
then let 
notice that 
so 
is cyclic and thus by the perpendiculars we get 
colinear, now let 
the A-excenter of 
, let 
the E,F excenters of 
respectively then finally let 
the D'-queue point of 
, notice how we have 
from the condition but also from radax 
colinear which forces that 
and 
and thus we are done by I-E Lemma.
midpoint of arc 
on 
then let 
notice that 
so 
is cyclic and thus by the perpendiculars we get 
colinear, now let 
the A-excenter of 
, let 
the E,F excenters of 
respectively then finally let 
the D'-queue point of 
, notice how we have 
from the condition but also from radax 
colinear which forces that 
and 
and thus we are done by I-E Lemma.
Z
K
Y
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#8
Apr 29, 2025, 11:59 PM
Y by
Sadigly wrote:
Probably from IMO SL, since this problem is also on Azerbaijan IMO TST, so please remove this post
No, it's not from IMO SL, but it's very curious where is it from then, given the coincidence.
Z
K
Y
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Y by
Assassino9931 wrote:
Sadigly wrote:
Probably from IMO SL, since this problem is also on Azerbaijan IMO TST, so please remove this post
No, it's not from IMO SL, but it's very curious where is it from then, given the coincidence.
What should we do with this problem — should we keep it or remove it? Problem 6 was similarly a duplicate and was removed before anyone wrote a solution. This one, however, has already been discussed, but I can still remove the statement if necessary — should I?
Z
K
Y
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N Quick Reply
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