IMO 2009 P2, but in space
by Miquel-point, May 14, 2025, 6:35 PM
Let 
be a tetrahedron with circumcenter 
. Suppose that the points 
and 
are interior points of the edges 
and 
, respectively. Let 
and 
be the centroids of the triangles 
, 
and 
, respectively. Prove that if the plane is tangent to the sphere 
then 
be a tetrahedron with circumcenter 
. Suppose that the points 
and 
are interior points of the edges 
and 
, respectively. Let 
and 
be the centroids of the triangles 
, 
and 
, respectively. Prove that if the plane is tangent to the sphere 
then 
Imtersecting two regular pentagons
by Miquel-point, May 14, 2025, 6:27 PM
The intersection of two congruent regular pentagons is a decagon with sides of 
in this order. Prove that
![\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]](//latex.artofproblemsolving.com/9/e/e/9ee73bbdc4b4f2cad2eb3fcfb3dbdf76b6200b4d.png)
in this order. Prove that![\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]](http://latex.artofproblemsolving.com/9/e/e/9ee73bbdc4b4f2cad2eb3fcfb3dbdf76b6200b4d.png)
Amazing projective stereometry
by Miquel-point, May 14, 2025, 6:24 PM
In the plane 
, given a circle 
and a point 
in its interior, not coinciding with the center of . Call a point of space, not lying on , a proper projection center if there exists a plane 
, not passing through , such that, by projecting the points of from to , the projection of is also a circle, and its center is the projection of . Show that the proper projection centers lie on a circle.
, given a circle 
and a point 
in its interior, not coinciding with the center of . Call a point of space, not lying on , a proper projection center if there exists a plane 
, not passing through , such that, by projecting the points of from to , the projection of is also a circle, and its center is the projection of . Show that the proper projection centers lie on a circle.Hard Inequality
by Asilbek777, May 14, 2025, 3:21 PM
Attachments:
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Proving radical axis through orthocenter
by azzam2912, May 14, 2025, 12:02 PM
In acute triangle 
let 
and 
denote the feet of the altitudes from 
and 
, respectively. Let line 
intersect circumcircle at points 
. Similarly, let line 
intersect circumcircle at points 
. Prove that the radical axis of circles 
and 
passes through the orthocenter of triangle
let 
and 
denote the feet of the altitudes from 
and 
, respectively. Let line 
intersect circumcircle at points 
. Similarly, let line 
intersect circumcircle at points 
. Prove that the radical axis of circles 
and 
passes through the orthocenter of triangle Old hard problem
by ItzsleepyXD, Apr 25, 2025, 4:15 AM
Let be a triangle and let be its circumcenter and 
its incenter.
Let be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .
Let
be the Gergonne point of the triangle .
Prove that line
is parallel with line 
.
be a triangle and let be its circumcenter and 
its incenter.Let
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .Let
be the Gergonne point of the triangle .Prove that line
is parallel with line 
.Easy Geometry
by pokmui9909, Mar 30, 2025, 5:18 AM
Triangle satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at , line 
meet 
at 
. Prove that the three lines 
are concurrent.
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at , line 
meet 
at 
. Prove that the three lines 
are concurrent.This post has been edited 1 time. Last edited by pokmui9909, Mar 30, 2025, 5:29 AM
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