be reals such that 
. Prove that
where 
are lengths of parallel sides and 
are other two sides.
and 
Prove that 
be a triangle. A circle passing through 
and 
intersects segments 
and 
at 
and 
,
and 
intersect at 
,
and 
intersect at 
.
if and only if 
.
Y by cubres
Given an acute triangle 
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
cyclic.
be the midpoint of 
where 
is the orthocenter, and let 
be the feet of the altitudes from 
.
is perpendicular to 
by symmetry, and as 
at 
by reims theorem, we have that 
.![\[\measuredangle PEQ=\measuredangle PED = \measuredangle PND=\measuredangle PAQ\]](http://latex.artofproblemsolving.com/2/8/5/285884435dc7a6bb542241f18bda104fe884c96e.png)
by parallel lines and nine point circle. This proves the claim.![\[\measuredangle PQD=\measuredangle PQE =\measuredangle PAE =\measuredangle EPA =\measuredangle EQA =\measuredangle DQH\]](http://latex.artofproblemsolving.com/7/e/5/7e5a8e580c89298a32268f32d79266e3a5b33c4d.png)
so 
bisects 
,
bisects 
as desired.
.
.
is concyclic.
.
bisects 
.
bisects 
.
,
.
