and 
.
is given. Let 
be the midpoint of 
,
be the foot of altitude from 
to 
be the intersection point of 
.
lies on the arc 
.
.
be a trapezium inscribed in a circle 
with diameter 
. The circle with center 
and radius 
meets 
and 
(where 
)
at 
.
is perpendicular to 
.
Y by Adventure10
Let 
be the point of the center of an ex-circle of the triangle 
, which touches the side 
. Let 
be the intersection point of the bisector of the angle 
of the triangle with the circumcircle of the triangle . Perpendicular from the point on the straight line 
, intersects the circumcircle of at the point 
. Prove, that if the points 
lie on the same line, then the triangle is isosceles.
(Mykola Moroz)
be the point of the center of an ex-circle of the triangle 
, which touches the side 
. Let 
be the intersection point of the bisector of the angle 
of the triangle with the circumcircle of the triangle . Perpendicular from the point on the straight line 
, intersects the circumcircle of at the point 
. Prove, that if the points 
lie on the same line, then the triangle is isosceles.(Mykola Moroz)
This post has been edited 5 times. Last edited by parmenides51, Jun 21, 2022, 1:38 AM
and 
.
as desired.
