and 
.
the center of the circumscribed circle of the triangle and 
the intersection of altitudes of this triangle. Line 
intersects 
in point 
and 
in point 
.
.
.
be a point on side 
be a point on the extension of 
such that 
.
,
satisfying 
.
of positive integers with 
prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
Y by Mango247, Mango247, Mango247
In triangle 
(
), the inscribed circle centered at point 
touches the side 
at point 
. Let 
be the midpoint of . Prove that the perpendiculars dropped from the points and on the lines 
and 
, respectively, intersect at the altitude of the triangle drawn from the vertex 
(or at its extension).
(
), the inscribed circle centered at point 
touches the side 
at point 
. Let 
be the midpoint of . Prove that the perpendiculars dropped from the points and on the lines 
and 
, respectively, intersect at the altitude of the triangle drawn from the vertex 
(or at its extension).This post has been edited 2 times. Last edited by parmenides51, Aug 29, 2020, 7:43 PM
the foot of the perpendicular from 
and let 
,
is the reflection of 
,
and 
,
,
we have 
,
is also tangent to 
,
all lie on a circle with diameter 
,
is cyclic, the. 
and 
hence the result.
