geometry problem with many circumcircles
by Melid, Jun 1, 2025, 5:32 AM
In scalene triangle 
, which doesn't have right angle, let 
be its circumcenter. Circle 
intersects 
and 
at 
and 
for the second time, respectively. Similarly, circle 
intersects 
and 
at 
and 
, and circle 
intersects 
and 
at 
and 
for the second time, respectively. Let 
and 
be circumcenters of triangle 
and 
, respectively. Prove that 
are collinear.
, which doesn't have right angle, let 
be its circumcenter. Circle 
intersects 
and 
at 
and 
for the second time, respectively. Similarly, circle 
intersects 
and 
at 
and 
, and circle 
intersects 
and 
at 
and 
for the second time, respectively. Let 
and 
be circumcenters of triangle 
and 
, respectively. Prove that 
are collinear.
Prove that
Prove that
Prove that
be an isosceles triangle with 
.
be a point on 
.
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
be a cyclic quadrilateral with 
being the midpoints of segments 
respectively. Suppose 
is the intersection of diagonals 
of quadrilateral 
Define 
to be the isogonal conjugate point of point 
Define 
similarly. Suppose 
intersects 
at a point 
Prove that: The Newton-Gauss line of quadrilateral 
satisfying 
and
for all 
.
of integers is rootiful if, for any positive integer 
and any 
,
are also in 
for positive integers 
and 
.
be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
April 2025
November 2024