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 .
(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
(3) Yufei Zhao, Lemmas in Euclidean Geometry