and 
.
of acute triangle 
is 
.
intersects 
at 
.
be 
.
,
,
,
,
,
similarly. Prove that 
,
,
are concurrent.
to 
(including 
(multiplication) between each number. Then he writes the rest of the numbers in the denominator and also writes 
.![\[\sqrt{a^2+1}+\sqrt{b^2+1}+\sqrt{c^2+1}\le \sqrt{2}(a+b+c)\]](http://latex.artofproblemsolving.com/6/e/d/6ed654c1d3616a8274ff0838e7bb13dfa0a11b8a.png)
Y by kaede_Arcadia, buratinogigle
We are posting the problems of the Synthetic Geometry Olympiad, which was recently concluded and hosted by kaede_Arcadia and myself.
Problem 1
Let
be a triangle with its 9-point center 
and excentral triangle 
. Denote the tangency points of the 
-excircle with sides 
, 
, and 
as 
, respectively. Similarly, define 
and 
for the 
- and 
-excircles.
Let
, 
, and 
. Let 
be the radical center of the circles 
, 
, and 
.
Prove that the lines
, 
, 
are concurrent.
Problem 2
Let be a triangle with circumcenter 
, incenter 
and incentral triangle 
. Let the line 
intersect 
again at 
. Similarly, define 
and 
.
Let
and 
be the 9-point centers of and 
, respectively.
Prove that the points
, 
are collinear.
Problem 3
Let be a triangle, and let 
be an isogonal conjugate pair. Suppose the line through 
and perpendicular to 
intersects 
again at 
. Similarly, define 
. Suppose the line through 
and perpendicular to 
intersects 
again at 
. Similarly, define 
.
Let
and 
be the orthocenters of 
and 
, respectively. Define 
and 
. Let 
and 
be the isogonal conjugates of and 
with respect to and , respectively.
Prove that the lines
are concurrent.
Problem 1
Let
be a triangle with its 9-point center 
and excentral triangle 
. Denote the tangency points of the 
-excircle with sides 
, 
, and 
as 
, respectively. Similarly, define 
and 
for the 
- and 
-excircles.Let
, 
, and 
. Let 
be the radical center of the circles 
, 
, and 
.Prove that the lines
, 
, 
are concurrent.Problem 2
Let
be a triangle with circumcenter 
, incenter 
and incentral triangle 
. Let the line 
intersect 
again at 
. Similarly, define 
and 
.Let
and 
be the 9-point centers of and 
, respectively.Prove that the points
, 
are collinear.Problem 3
Let
be a triangle, and let 
be an isogonal conjugate pair. Suppose the line through 
and perpendicular to 
intersects 
again at 
. Similarly, define 
. Suppose the line through 
and perpendicular to 
intersects 
again at 
. Similarly, define 
.Let
and 
be the orthocenters of 
and 
, respectively. Define 
and 
. Let 
and 
be the isogonal conjugates of and 
with respect to and , respectively.Prove that the lines
are concurrent.This post has been edited 1 time. Last edited by kooooo, Feb 15, 2025, 11:35 AM
Reason: Typo
Reason: Typo
