Geo practice
by MP8148, Jul 24, 2024, 9:03 PM
I tried some ISL geometry problems:
I tried G5 for a while but it looked kinda hard so I gave up. I didn't look at the rest of the problems.
Also, I passed the driver's test somehow, even though I'm horrible driver and I drive into a wall recently while parking
G1 wrote:
Let 
be a convex pentagon such that 
. Suppose that the midpoint of 
is the circumcenter of triangle 
. Let 
be the circumcenter of triangle 
.
Prove that line
passes through the midpoint of segment 
.
be a convex pentagon such that 
. Suppose that the midpoint of 
is the circumcenter of triangle 
. Let 
be the circumcenter of triangle 
.Prove that line
passes through the midpoint of segment 
.
,
,
is a parallelogram.
as unit circle, and decided to give up and look at the solutions.G2 wrote:
Let 
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from to 
. Ray 
mets again at 
. Point 
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
. Prove that lies on .
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from to 
. Ray 
mets again at 
. Point 
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
. Prove that lies on .
so 
,
.
by angle chase, which kills the problem.G3 wrote:
Let 
be a cyclic quadrilateral with 
. Let 
be the midpoint of the arc not containing 
. Suppose there is a point inside such that 
and 
.
Prove that lines
, and 
are concurrent.
be a cyclic quadrilateral with 
. Let 
be the midpoint of the arc not containing 
. Suppose there is a point inside such that 
and 
.Prove that lines
, and 
are concurrent.
,
to meet 
at 
,
,
.
,
,
and 
are homothetic. The center of the homothety is 
which lies on 
,G4 wrote:
Let be an acute-angled triangle with 
. Let 
be the circumcircle of . Let be the midpoint of the arc 
of containing . The perpendicular from to meets 
at and meets again at 
. The line through parallel to meets line at 
. Denote the circumcircle of triangle 
by . Let meet again at 
. Prove that the line tangent to at meets line on the internal angle bisector of 
.
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let be the midpoint of the arc 
of containing . The perpendicular from to meets 
at and meets again at 
. The line through parallel to meets line at 
. Denote the circumcircle of triangle 
by . Let meet again at 
. Prove that the line tangent to at meets line on the internal angle bisector of 
.
.
where 
is the point on 
.
and 
are homothetic, and that finishes the problem.I tried G5 for a while but it looked kinda hard so I gave up. I didn't look at the rest of the problems.
Also, I passed the driver's test somehow, even though I'm horrible driver and I drive into a wall recently while parking
March 2025
July 2024