Y by Adventure10, Mango247
Let ABC be a triangle and its incircle touches BC,CA,AB at D,E, F, reps. AD cuts (I) again at G. H lies on line EF such that 
. Prove that 
.
. Prove that 
.G
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Sequence of projections is convergent
Filipjack 0
Source: Romanian National Olympiad 1997 - Grade 10 - Problem 3
A point 
and two lines 
and 
are given in the space. For each nonnegative integer 
we denote by 
the projection of 
on 
and by 
the projection of on 
Prove that there exist two segments ![$[A'A''] \subset d_1$](//latex.artofproblemsolving.com/8/8/6/88667ee717ddd630efd34a9c4c83ee359c4897ed.png)
and ![$[B'B''] \subset d_2$](//latex.artofproblemsolving.com/3/7/4/374927492266c8adaed5d34e3871d250baba9c1b.png)
of length 
and a nonnegative integer 
such that ![$A_n \in [A'A'']$](//latex.artofproblemsolving.com/4/8/5/4851110dd8f0bf0ee7e42239ee051da76f5fbd65.png)
and ![$B_n \in [B'B'']$](//latex.artofproblemsolving.com/5/e/7/5e7d78dd940f5c81673b779a9b45e83f4dc05b4d.png)
for any 
and two lines 
and 
are given in the space. For each nonnegative integer 
we denote by 
the projection of 
on 
and by 
the projection of on 
Prove that there exist two segments ![$[A'A''] \subset d_1$](http://latex.artofproblemsolving.com/8/8/6/88667ee717ddd630efd34a9c4c83ee359c4897ed.png)
and ![$[B'B''] \subset d_2$](http://latex.artofproblemsolving.com/3/7/4/374927492266c8adaed5d34e3871d250baba9c1b.png)
of length 
and a nonnegative integer 
such that ![$A_n \in [A'A'']$](http://latex.artofproblemsolving.com/4/8/5/4851110dd8f0bf0ee7e42239ee051da76f5fbd65.png)
and ![$B_n \in [B'B'']$](http://latex.artofproblemsolving.com/5/e/7/5e7d78dd940f5c81673b779a9b45e83f4dc05b4d.png)
for any 
0 replies
Right-angled triangle if circumcentre is on circle
liberator 76
N
by numbertheory97
Source: IMO 2013 Problem 3
Let the excircle of triangle 
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
Proposed by Alexander A. Polyansky, Russia
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.Proposed by Alexander A. Polyansky, Russia
76 replies
APMO 2016: Great triangle
shinichiman 26
N
by ray66
Source: APMO 2016, problem 1
We say that a triangle is great if the following holds: for any point 
on the side , if 
and 
are the feet of the perpendiculars from to the lines and 
, respectively, then the reflection of in the line 
lies on the circumcircle of the triangle . Prove that triangle is great if and only if 
and 
.
Senior Problems Committee of the Australian Mathematical Olympiad Committee
is great if the following holds: for any point 
on the side , if 
and 
are the feet of the perpendiculars from to the lines and 
, respectively, then the reflection of in the line 
lies on the circumcircle of the triangle . Prove that triangle is great if and only if 
and 
.Senior Problems Committee of the Australian Mathematical Olympiad Committee
26 replies
IMO ShortList 2001, geometry problem 2
orl 48
N
by legogubbe
Source: IMO ShortList 2001, geometry problem 2
Consider an acute-angled triangle . Let be the foot of the altitude of triangle issuing from the vertex , and let 
be the circumcenter of triangle . Assume that 
. Prove that 
.
. Let be the foot of the altitude of triangle issuing from the vertex , and let 
be the circumcenter of triangle . Assume that 
. Prove that 
.
48 replies
Projective Electrostatistics
drago.7437 0
Source: Own
Given two charges of any magnitude , a third charge collinear with them , exists such that it is in equillibirum , Prove that if a fourth charge in the same line exists such that it is in equillibrium , then the 3rd charge and the fourth charge are harmonic conjugates with respect to the two fixed charges . , For example if two +q charges are fixed then if in their midpoint placed a charge -q , it is in equillibrium , also if the same charge -q is placed at infinity the system is again in equillibrium , and the midpoint and the point at infinity are harmonic conjugates .
0 replies
Find an angle
socrates 3
N
by Nari_Tom
Source: Baltic Way 2016, Problem 18
Let 
be a parallelogram such that 
Let 
and 
be the midpoints of and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
be a parallelogram such that 
Let 
and 
be the midpoints of and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
3 replies
Two Orthocenters and an Invariant Point
Mathdreams 1
N
by RANDOM__USER
Source: 2025 Nepal Mock TST Day 1 Problem 3
Let 
be a triangle, and let be an arbitrary point on line 
, where is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of .
(Kritesh Dhakal, Nepal)
be a triangle, and let be an arbitrary point on line 
, where is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of .(Kritesh Dhakal, Nepal)
1 reply
Geometry
youochange 2
N
by youochange
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 
.
2 replies
Beautiful problem
luutrongphuc 20
N
by r7di048hd3wwd3o3w58q
Let triangle be circumscribed about circle 
, and let 
be the orthocenter of . The circle touches line at . The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
be circumscribed about circle 
, and let 
be the orthocenter of . The circle touches line at . The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
20 replies
Common tangent to diameter circles
Stuttgarden 1
N
by jrpartty
Source: Spain MO 2025 P2
The cyclic quadrilateral , inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals and 
. The circle with center and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals and 
. The circle with center and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
1 reply
Ratios in a right triangle
PNT 1
N
by Mathzeus1024
Source: Own.
Let be a right triangle in with 
. Let be the midpoint of and a point on such that 
. Let 
.
Compute in terms of
and the ratio 
.
be a right triangle in with 
. Let be the midpoint of and a point on such that 
. Let 
.Compute in terms of
and the ratio 
.
1 reply
parallel
G
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G
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Source: unsolved geometry
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Y by Adventure10
Dear Mathlinkers,
have a look at
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=111468
Sincerely
Jean-Louis
have a look at
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=111468
Sincerely
Jean-Louis
Z
K
Y
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N Quick Reply
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