In this post I wanted to show a couple of similar problems that can be proved using some ideas from projective geometry.
The first problem I found via the Romantics of Geometry Facebook group: let 
be the point of tangency of the incircle of 
with 
and let 
be the foot of the perpendicular from the incentre 
of the to 
. Then show 
bisects 
.
The second problem is motivated by the above and problem 2 of the 2008 USAMO: this time let be a symmedian of 
and be the foot of the perpendicular from the circumcentre of to . Then show that bisects .
Here is a solution to the first problem inspired bythat of Vaggelis Stamatiadis. Let the line through the other two points of tangency 
of the incircle with intersect line at the point 
as shown below. Note that since 
and 
are tangents to the circle, line 
is the polar of 
with respect to the incircle.
Since is on the polar of , by La Hire’s theorem, is on the polar of . The polar of also passes through (as 
is a tangent to the circle at ). We conclude that the polar of is the line through and .
Next, let 
intersect 
at 
. By theorem 5(a) at this link, the points 
form a harmonic range. Since the cross ratio of collinear points does not change under central projection, considering the projection from , 
also form a harmonic range. (Alternatively, this follows from the theorems of Ceva and Menelaus using the Cevians intersecting at the Gergonne point and transveral ). Also, 
as both 
and 
are perpendicular to polar of .
Considering a central projection from of line 
to a line 
parallel to 
through , we see that 
form a harmonic range. Since 
is a point at infinity, this implies is the midpoint of 
and so triangles 
and 
are congruent (equality of two pairs of sides and included angle is 
). Hence bisects as was to be shown.
For the second problem, we use the following characterisation of a symmedian: extended concurs with the lines of tangency of the circumcircle at 
and 
. (For three proofs of this see here.)
Define as the intersection of 
with and 
as the intersection of with the tangents at 
. Note that line 
is the polar of with respect to the circumcircle. By La Hire’s theorem, must be on the polar of . This polar is perpendicular to 
(the line joining to the centre of the circle) and as 
by construction of , it follows that line 
is the polar of . Again by theorem 5(a) in reference (2), 
form a harmonic range. Following the same argument as the previous proof, this together with imply bisects as required.
By similar arguments, one can prove the following, left to the interested reader. If is the -excentre of , the ex-circle’s point of tangency of , and the foot of the perpendicular from to line , then bisects .
References
(1) Alexander Bogomolny, Poles and Polars from Interactive Mathematics Miscellany and Puzzles http://www.cut-the-knot.org/Curriculum/Geometry/PolePolar.shtml, Accessed 19 March 2017
(2) Poles and Polars – Another Useful Tool! | The Problem Solver’s Paradise
(3) Yufei Zhao, Lemmas in Euclidean Geometry
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