Given is a triangle ABC and let D be the midpoint the major arc BAC of its circumcircle. Let M,N be the midpoints of AB,AC and J,E,F are the touchpoints of the incircle (I) of △ABC with BC,CA,AB. The line MN intersects JE,JF at K,H respectively; IJ intersects the circle (BIC) at G and DG intersects (BIC) at T. a) Prove that JA passes through the midpoint of HK and is perpendicular to IT. b) Let R,S respectively be the perpendicular projection of D on AB,AC. Take the points P,Q on IF,IE respectively such that KP and HQ are both perpendicular to MN. Prove that the three lines MP,NQ and RS are concurrent .
Problem
Source: VMO 2023 day 1 P4
Tags: geometry
24.02.2023 20:37
a_507_bc wrote: JA intersects the circle (BIC) at G Is it correct? I think it must be JA intersects the circle (ABC) at G.
24.02.2023 20:59
It is IJ, sorry for the typo
24.02.2023 21:00
First part of a) JA passes through the midpoint of HK is just Iran Lemma. We know that B,I,K are collinear and ∠BKA=90∘, so ∠AKH=90∘−∠HKB=90∘−12∠B=∠FJB=∠JHK, so AK and HJ are parallel. And because MN bisects AJ, then AJ bisect HK. (AHJK is a parallelogram).
24.02.2023 22:59
25.02.2023 08:21
this geo easy for a VMO's
25.02.2023 10:22
Some good looking properties (May be c)) Prove that: PQ⊥IZ with Z midpoint BC Prove that: PQ bisects IZ. (Collected)
25.02.2023 18:40
Way too many points to look at Part a is easy For part b, the lines concurrent at Spieker Center
25.02.2023 18:44
Well, in recent years VMO geometry problems aren't good. Part a and b are independent of each other, and the committee just use old ideas to create these unsatisfying geo
25.02.2023 18:47
wardtnt1234 wrote: Well, in recent years VMO geometry problems aren't good. Part a and b are independent of each other, and the committee just use old ideas to create these unsatisfying geo Yes, you are right. We just need 1 high quality and good problem in the test more than 2 independent problems but not very nice and complicated. There are still more issues about this but I will just say about the Geometry...
10.07.2023 15:49
Parts a. and b. are solved in separate images because they are not related to each other.