This blog post corresponds to my newest olympiad handout on mixtilinear incircles.
My favorite circle associated to a triangle is the 
Here’s the picture:
The points 
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
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
- An Iran 2002 problem asks to show that ray
passes through the point diametrically opposite
is a symmedian of the triangle
and (by the previous fact) that
- Lines
and
are isogonal. This was essentially EGMO 2012, Problem 5, and the “morally correct” solution is to do an inversion at
-bisector (sometimes we call this a “
inversion”).
- As a consequence of this, one can also show that lines
are isogonal (with respect to
).
- One can also deduce from this that the circumcircle of
passes through the intersection of
.
- As a consequence of this, one can also show that lines
- Lines
and
- Moreover, line
To put that all into one picture:
Power Overwhelming 
#2 is also a key in solving 2014 ISL G7.
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A wonderful article indeed.Just to contribute an idea that occurred to me just now I think much more could be proved via square root bc inversion.For example if instead there was a mixitilinear excircle opposite A and we were to prove that its touch chord passes through the A excenter the above mentioned technique gives a neat proof.
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Something more came to me, both properties 1 and 2 are a direct consequence of $\sqrtbc$ inversion.This might add a little bit of insight into this already wonderful article.This technique somehow naturally connects the mixitilinear incircle with the excircle and make things easier a bit.Of course its just an idea that i think might add something more to the list.
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Yep, sqrt bc inverison will certainly prove the other properties here. :) The trick is that the mixtilinear incircle gets swapped with the excircle, which is much easier to deal with.
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Awesome! I was really wondering how to prove some of these properties in class. Btw really nice blog!
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Well, my favorite circle is the B-mixtillinear circle ^_^
On a more serious note, perhaps you could also include that the radical axis of the B,C mixt. circles goes through the midpoint of arc $BC$ and the midpoint of $ID$, where $D$ is the tangency point of the incircle with $BC$. It’s not very useful, but it’s nice to know it exists.
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