2019 Poland Math Olympiad Geometry problem (Symmedian line, Incenter Excenter Lemma, Monge Theorem)

2019 Polish Math Olympiad

[tmasacademy]

The God has started to post once again! Coach Nal is back in youtube business.

[andreivila]

Great problem sir! I found a quicker solution tho... Like you proved, P is the center of homothety between the incircle of AKL and the circumcircle of ABC. Clearly A is the center of homothety between the circumcircle of ABC and the circumcircle of AKL (that follows from the fact that AK and AL are isogonal), so H, the center of homothety between the incircle and circumcircle of AKL, is on AP. But it’s a known fact (which is again derived from homotheties) that the tangency point of the A-mixtllinear circle and the circumcircle of AKL is on the line AH. So A, P, H are collinear. But AH and AE are isogonal (preform symetric inversion about A in triangle AKL) so the conclusion follows from here. Have a great day! :)

P.S. It is an honor to have such a legend return! Keep up the good work!!

[tndkpgp]

Beautiful application of Monge's theorem (preparing for USAMO :D).

[pranav_]

Osman sir
Please don't stop it again 🥺

[rustemtehmezov]

Great problem and great solution! (But i just don't understand why P is excimilicenter of big omega and qamma)

[tonyha]

Thanks for a very good lecture, I learn a lot from it. But I have 2 questions. 1. @16.12 we have OB=OD=OC then D is the incenter of trianlge BQC, why? Thanks in advance!

[helo]

Hi Osman, is it okay if you do 2017 USAJMO Problems 2 and 4?

[davidepierrat]

Double monge ftw. Such a nice theorem if you ask me :D

[Rex-fevl]

Where did you learn how to write proofs and what type do you use?