be a triangle with incenter 
,
and 
at 
and 
.
and 
the reflections of 
be the intersection of 
with 
,
be the midpoint of 
and 
are perpendicular.
that verify the following equation for all 
:
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
be the circle with diameter 
.
be the intersection of 
and 
.
be the intersection of 
and 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
that satisfy the equation 
.
intersect 
Circles 
are internally tangent to 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
Y by Mango247
A circle and a point 
inside it are given. Two arbitrary perpendicular rays starting at point intersect the circle at points 
and 
. Point 
is the projection of point onto line 
is the intersection point of tangents to the circle drawn through points and . Prove that all lines 
pass through the same point.
(A. Zaslavsky)
inside it are given. Two arbitrary perpendicular rays starting at point intersect the circle at points 
and 
. Point 
is the projection of point onto line 
is the intersection point of tangents to the circle drawn through points and . Prove that all lines 
pass through the same point.(A. Zaslavsky)
This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 2:03 AM
and radius 
.
and 
.
;
.
.
lies on the line 
.
