.
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
be a triangle whose incircle is tangent to 
,
,
at 
,
and 
be the midpoints of 
and 
,
and 
lie on 
so that 
and 
.
and 
intersect on the circumcircle of 
.
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
Y by
In the tetrahedron 
the plane angles 
, 
and 
are equal to each other. The points 
, 
and 
, are selected on sides 
and 
respectively, that triangles 
, 
and 
have the smallest possible perimeters. Prove that lines 
, 
and 
intersect at one point.
(M. Plotnikov)
the plane angles 
, 
and 
are equal to each other. The points 
, 
and 
, are selected on sides 
and 
respectively, that triangles 
, 
and 
have the smallest possible perimeters. Prove that lines 
, 
and 
intersect at one point.(M. Plotnikov)
