So I was doing USEMO 2022 P4, and I ended up dying a great amount of time on it. But while typing the solution, I noticed that the configurations can be incredibly extended, and upto such an extent, that I felt it wont be appropriate to post such a long post on the problem thread. So here we go . Also, I believe that the configuration can be extended wayyyyy more than the current extension due to the close resemblance to the Brokard's Theorem which I am too nub to continue further expedition on .

We conclude the following observations from this post:

• are to each other.
• are to each other.
• The pairs of the two triplets and are to each other.
• are collinear.
• are collinear.

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We begin off with a moderate level Sub-Problem .


Proof: Simple angle chasing follows (yeah yeah, in other words, left for the reader to prove ).

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Now we have a little taste of the actual problem .


Proof: Let . Now, is tangent to . Similarly, is tangent to . Thus, .

Now, . A similar argument gives from which we can conclude that is infact a kite.

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Now we move onto solving the actual problem .

We use the labellings as specified in the figure (too many points to latex labellings lol). Let be the angle bisector of and be the angle bisector of .



From our subproblem, we have but is a kite . Similarly and . But from here, we also had that , thus we can wrap up all our present information into the following, and are two triplets where pairs from the same triplet are to each other and pairs of different triplet are to each other.

Finally, Pappus on and gives are collinear. And another Pappus on and gives are collinear.

And yeah thats all !