Let me study those curves of yours
by shiningsunnyday, Oct 30, 2016, 4:24 AM
Euclidean Geometry in Mathematical Olympiads wrote:
Let
be a triangle and
be a point on
. Suppose a circle
is tangent to
at
at
, and also to
Then the incenter of
lies on line 
Note that the circle is better-known as a curvilinear incircle of
a special case of which is the A-mixtilinear incircle.










Note that the circle is better-known as a curvilinear incircle of

The idea is to let
be the intersection of
and
and prove it's the incircle via the incenter-excenter lemma. Specifically, it suffices to prove 
Lemma 1:
Proof: It suffices to show
which follows from 
Now we desire
But direct angle chasing is futile until we establish another intermediate lemma.
Lemma 2:
is cyclic.
Proof:
This will allow us to finish.
Lemma 3:
Let's work backwards from the cute observation that
is isosceles:
![\[\angle{DLK}=\angle{DKL} \]](//latex.artofproblemsolving.com/3/f/e/3fe5433e93f3c09ec935c93fe2448d975f6f1ade.png)
![\[\angle{ILC} = \angle{LTK} \]](//latex.artofproblemsolving.com/d/5/1/d51b571d81f505ff3c1e7fee2bdf5c7afebd0188.png)
![\[\angle{ITC} = \angle{LTK} \]](//latex.artofproblemsolving.com/8/1/b/81b0ba230dab71d314555c724b4fb32d1567ab5d.png)
![\[\angle{ITC} + \angle{ITL} = \angle{LTK} + \angle{ITL} \]](//latex.artofproblemsolving.com/5/7/d/57dfdd7c1010b70e878010abd1ddd82f37c09655.png)
![\[\angle{CTL} = \angle{ITK}\]](//latex.artofproblemsolving.com/7/4/9/749e8ae09a633cab27d0ba25bb5ffd567d619545.png)
where we used Lemma 2 in lines three and six. The last line finishes the lemma and thus
which combined with the incenter-excenter lemma and lemma 1 finishes the problem.




Lemma 1:

Proof: It suffices to show


Now we desire

Lemma 2:

Proof:

This will allow us to finish.
Lemma 3:

Let's work backwards from the cute observation that

![\[\angle{DLK}=\angle{DKL} \]](http://latex.artofproblemsolving.com/3/f/e/3fe5433e93f3c09ec935c93fe2448d975f6f1ade.png)
![\[\angle{ILC} = \angle{LTK} \]](http://latex.artofproblemsolving.com/d/5/1/d51b571d81f505ff3c1e7fee2bdf5c7afebd0188.png)
![\[\angle{ITC} = \angle{LTK} \]](http://latex.artofproblemsolving.com/8/1/b/81b0ba230dab71d314555c724b4fb32d1567ab5d.png)
![\[\angle{ITC} + \angle{ITL} = \angle{LTK} + \angle{ITL} \]](http://latex.artofproblemsolving.com/5/7/d/57dfdd7c1010b70e878010abd1ddd82f37c09655.png)
![\[\angle{CTL} = \angle{ITK}\]](http://latex.artofproblemsolving.com/7/4/9/749e8ae09a633cab27d0ba25bb5ffd567d619545.png)
![\[\angle{MIK} = \angle{ITK}\]](http://latex.artofproblemsolving.com/9/c/e/9ce6a4273c67d5837b61c99d062beecbda0a39be.png)

Tidbit
Woohoo currently learning curvilinear and mixtilinear incircles in EGMO and Lemmas. Also got a near-perfect index on a practice PSAT today -> national merit (the official one's on Wed), so life's not that bad.
But...
But...