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
be an acute triangle with 
,
its orthocenter and 
be a point on the parallel through 
to 
and 
are on different sides of 
the intersection of the circumcircle of 
,
is the reflection of 
the midpoint of 
,
the intersection of 
and the parallel through 
,
the intersection of 
and the perpendicular through 
and 
a point on 
,
is the midpoint of 
.
lie on a line.
i'll probably post 2 more problems related to it soon, i'd say they're easier than this though
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 all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
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 
.
and any 
,
are also in 
for positive integers 
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
September 2017
February 2017