such that for all 
we have:
be an acute triangle inscribed in its circumcircle 
,
be its incircle. Let 
be the point where the 
incircle of triangle 
touches 
.
intersects segment 
at 
,
at 
.
perpendicular to 
at 
.![\[AI \parallel PA'.\]](http://latex.artofproblemsolving.com/0/2/6/0264e64b5edff4aca1e35037c5b0f86763c3b4e7.png)
in its exterio
so that 
,
,
.
and
Y by Adventure10, Mango247
For any point 
on a segment 
, isosceles and right-angled triangles 
and 
are constructed on the same side of , with 
and 
as the bases. Determine the locus of the midpoint 
of 
when describes the segment .
on a segment 
, isosceles and right-angled triangles 
and 
are constructed on the same side of , with 
and 
as the bases. Determine the locus of the midpoint 
of 
when describes the segment .
