be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
be an acute triangle. Let 
be a point on side 
and 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
,
and 
.
and 
intersect at a point 
.
,
are collinear.
.
Y by Adventure10, Mango247
Points 
and 
are the midpoints of the sides 
and 
of the triangle 
, 
is the center of its inscribed circle, 
is the point of tangency of the inscribed circles with side 
. The external angle bisector at the vertex 
intersects the line 
at the point 
, and the external angle bisector at the vertex of 
intersects at 
. Prove that the area of the quadrilateral 
is equal to half the area of the triangle .
and 
are the midpoints of the sides 
and 
of the triangle 
, 
is the center of its inscribed circle, 
is the point of tangency of the inscribed circles with side 
. The external angle bisector at the vertex 
intersects the line 
at the point 
, and the external angle bisector at the vertex of 
intersects at 
. Prove that the area of the quadrilateral 
is equal to half the area of the triangle .
