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Simson lines on OH circle
DVDTSB 3
N
4 hours ago
by Funcshun840
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







Prove that the orthocenter of


Note. Assume that the points

Proposed by Andrei Chiriță
3 replies
IMO Shortlist 2012, Geometry 8
lyukhson 33
N
5 hours ago
by awesomeming327.
Source: IMO Shortlist 2012, Geometry 8
Let
be a triangle with circumcircle
and
a line without common points with
. Denote by
the foot of the perpendicular from the center of
to
. The side-lines
intersect
at the points
different from
. Prove that the circumcircles of the triangles
,
and
have a common point different from
or are mutually tangent at
.
Proposed by Cosmin Pohoata, Romania
















Proposed by Cosmin Pohoata, Romania
33 replies
IMO 2012 P5
mathmdmb 123
N
6 hours ago
by SimplisticFormulas
Source: IMO 2012 P5
Let
be a triangle with
, and let
be the foot of the altitude from
. Let
be a point in the interior of the segment
. Let
be the point on the segment
such that
. Similarly, let
be the point on the segment
such that
. Let
be the point of intersection of
and
.
Show that
.
Proposed by Josef Tkadlec, Czech Republic















Show that

Proposed by Josef Tkadlec, Czech Republic
123 replies
Fixed line
TheUltimate123 14
N
6 hours ago
by amirhsz
Source: ELMO Shortlist 2023 G4
Let
be a point on segment
. Let
be a fixed circle passing through
, and let
be a variable point on
. Let
be the intersection of the tangent to the circumcircle of
at
and the tangent to the circumcircle of
at
. Show that as
varies,
lies on a fixed line.
Proposed by Elliott Liu and Anthony Wang













Proposed by Elliott Liu and Anthony Wang
14 replies
KJMO 2001 P1
RL_parkgong_0106 1
N
Today at 2:05 PM
by JH_K2IMO
Source: KJMO 2001
A right triangle of the following condition is given: the three side lengths are all positive integers and the length of the shortest segment is
. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.

1 reply
Cotangential circels
CountingSimplex 5
N
Today at 2:02 PM
by rong2020
Let
be a triangle with circumcenter
and let the angle bisector of
intersect 
at
. The point
is such that
and
. Lines
and
intersect at
the point
. Show that the circle centered at
and passing through
is tangent to segment
.




at






the point




5 replies
Midpoint in a weird configuration
Gimbrint 0
Today at 1:39 PM
Source: Own
Let
be an acute triangle (
) with circumcircle
. Point
is chosen on arc
, not containing
, so that, letting
intersect
at
, one has
. Points
and
lie on lines
and
respectively, such that
is a parallelogram. Point
is chosen on arc
, not containing
, such that
. Line
intersects
at
, and line
intersects
at
. Line
intersects
,
and
at points
,
and
respectively.
Prove that
.
































Prove that

0 replies
Quadrangle, nine-point conic, Steiner line
kosmonauten3114 0
Today at 1:23 PM
Source: My own
Let
be a general quadrangle which does not form an orthocentric system. Let
,
,
,
be the Euler-Poncelet point (
), isogonal center (
), midray homothetic center (
), inscribed square axes crosspoint (
) of
, respectively.
Let
be the orthocenter of
, and define
,
,
cyclically.
Let
(
).
Let
(
).
Then, the 12 points
,
,
,
,
,
,
,
,
,
,
,
lie on the same conic, here denoted by
.
Let
be the nine-point conic of
.
Suppose that
and
have 4 distinct real intersection points, and let
,
,
be the intersections, other than
, of
and
.
Prove that the Steiner line of
with respect to
passes through
and
, and show that the center of
and the orthocenter of
coincide with
.










Let





Let


Let


Then, the 12 points













Let


Suppose that








Prove that the Steiner line of







0 replies
My Unsolved Problem
ZeltaQN2008 1
N
Today at 12:58 PM
by Funcshun840
Source: IDK
Let triangle
be inscribed in circle
. Let
be the
-excircle of triangle
, which is tangent to
, the extension of
, and the extension of
. Let
and
be the angle bisectors of triangle
. Let
intersect
at two points
and
.
a) Prove that circle
bisects the segments
and
.
b) Prove that
and
are the points of tangency of the common external tangents of circles
and
.















a) Prove that circle



b) Prove that




1 reply

Geometry Concurrence
KHOMNYO2 1
N
Today at 12:32 PM
by Funcshun840
Given triangle
such that
. Let excircle-
be tangent with
at points
respectively. Let
and
be points on the segment
respectively such that
is parallel to
. Lastly, let
be the circle that is externally tangent with the excircle-
on point
. Prove that
, and
concur at a point.















1 reply
