orthocenter on sus circle
by DVDTSB, May 13, 2025, 12:18 PM
Let 
be an acute triangle with 
, and let 
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
. The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point 
. Let 
be a point on the ray 
, starting at , such that 
.
Show that the orthocenter of triangle lies on the circle with diameter 
.
Proposed by Radu Lecoiu
be an acute triangle with 
, and let 
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
. The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point 
. Let 
be a point on the ray 
, starting at , such that 
.Show that the orthocenter of triangle
lies on the circle with diameter 
.Proposed by Radu Lecoiu
grid contains a number from 
to 
;
that satisfies the following condition:
be two triangles inscribed in the same circle, centered at 
.
with respect to triangle 
.
.
lie in this order on the circle and form a convex, non-degenerate hexagon.
and 
Show that
and 
Show that
Let 
Show that
and let 
be positive integers such that 
.
and determine when equality holds.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
,
,
at point 
.
,
,
,
,
can be divided into two groups, so that the angles in each group have a sum of 
.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
that satisfies 
for all real 
May 2020
January 2020