MathDB

2

Part of 2022 Iranian Geometry Olympiad

Problems(3)

Isosceles Trapezoid

Source: IGO 2022 Elementary P2

12/14/2022
An isosceles trapezoid ABCDABCD (AB∥CD)(AB \parallel CD) is given. Points EE and FF lie on the sides BCBC and ADAD, and the points MM and NN lie on the segment EFEF such that DF=BEDF = BE and FM=NEFM = NE. Let KK and LL be the foot of perpendicular lines from MM and NN to ABAB and CDCD, respectively. Prove that EKFLEKFL is a parallelogram.
Proposed by Mahdi Etesamifard
trapezoidgeometryparallelogramiranian geometry olympiad
Two congruent circles and parallel lines

Source: IGO 2022 Intermediate P2

12/13/2022
Two circles ω1\omega_1 and ω2\omega_2 with equal radius intersect at two points EE and XX. Arbitrary points C,DC, D lie on ω1,ω2\omega_1, \omega_2. Parallel lines to XC,XDXC, XD from EE intersect ω2,ω1\omega_2, \omega_1 at A,BA, B, respectively. Suppose that CDCD intersect ω1,ω2\omega_1, \omega_2 again at P,QP, Q, respectively. Prove that ABPQABPQ is cyclic.
Proposed by Ali Zamani
geometryhomothety
IGO 2022 advanced/free P2

Source: Iranian Geometry Olympiad 2022 P2 Advanced, Free

12/13/2022
We are given an acute triangle ABCABC with AB≠ACAB\neq AC. Let DD be a point of BCBC such that DADA is tangent to the circumcircle of ABCABC. Let EE and FF be the circumcenters of triangles ABDABD and ACDACD, respectively, and let MM be the midpoints EFEF. Prove that the line tangent to the circumcircle of AMDAMD through DD is also tangent to the circumcircle of ABCABC.
Proposed by Patrik Bak, Slovakia
geometry