be a cyclic quadrilateral. Let 
,
be the feet of the perpendiculars from 
to the lines 
,
,
,
if and only if the bisectors of 
and 
are concurrent with 
.
be reals such that 
. Prove that
be the feet of the altitudes from 
to 
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
where 
are lengths of parallel sides and 
are other two sides.
Y by XbenX, Adventure10, Mango247
Let 
be a triangle with an inscribed circle centered at 
. The line perpendicular to 
at intersects 
at 
such that, 
lies closer to 
than 
. Let 
. Prove that, 
is the angle bisector of 
be a triangle with an inscribed circle centered at 
. The line perpendicular to 
at intersects 
at 
such that, 
lies closer to 
than 
. Let 
. Prove that, 
is the angle bisector of 
This post has been edited 1 time. Last edited by AlastorMoody, Mar 30, 2019, 12:09 PM
.
be the intersection of 
with 
we see that 
.
is cyclic 
is an isosceles trapezium. So 
.
.
be the intersection of 
respectively.Clearly 
cyclic trapeziod
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be the intersection of 
. We need to show 
.
and we need to show 
we have 
and from the well known fact that 
we have 
