This blog post corresponds to my newest olympiad handout on mixtilinear incircles.
My favorite circle associated to a triangle is the -mixtilinear incircle. While it rarely shows up on olympiads, it is one of the richest configurations I have seen, with many unexpected coincidences showing up, and I would be overjoyed if they become fashionable within the coming years.
Here’s the picture:
The points and are the contact points of the incircle and -excircle on the side . Points , , are the midpoints of the arcs.
As a challenge to my recent USAMO class (I taught at A* Summer Camp this year), I asked them to find as many “coincidences” in the picture as I could (just to illustrate the richness of the configuration). I invite you to do the same with the picture above.
The results of this exercise were somewhat surprising. Firstly, I found out that students without significant olympiad experience can’t “see” cyclic quadrilaterals in a picture. Through lots of training I’ve gained the ability to notice, with some accuracy, when four points in a diagram are concyclic. This has taken me a long way both in setting problems and solving them. (Aside: I wonder if it might be possible to train this skill by e.g. designing an “eyeballing” game with real olympiad problems. I would totally like to make this happen.)
The other two things that happened: one, I discovered one new property while preparing the handout, and two, a student found yet another property which I hadn’t known to be true before. In any case, I ended up covering the board in plenty of ink.
Here’s the list of properties I have.
- First, the classic: by Pascal’s Theorem on , we find that points , , are collinear; hence the contact chord of the -mixtilinear incircle passes through the incenter. The special case of this problem with appeared in IMO 1978.
- Then, by Pascal on , we discover that lines , , and are also concurrent.
- This also lets us establish (by angle chasing) that and are concyclic. In addition, lines and are tangents to these circumcircles at (again by angle chasing).
- As a consequence of this, one can also show that lines and are isogonal (with respect to ).
- One can also deduce from this that the circumcircle of passes through the intersection of and .
To put that all into one picture: