MathDB

2

Part of 2019 South East Mathematical Olympiad

Problems(2)

Two circles are congruent if tangent

Source: CSMO 2019 Grade 10 Problem 2

7/30/2019
Two circles Γ1\Gamma_1 and Γ2\Gamma_2 intersect at A,BA,B. Points C,DC,D lie on Γ1\Gamma_1, points E,FE,F lie on Γ2\Gamma_2 such that A,BA,B lies on segments CE,DFCE,DF respectively and segments CE,DFCE,DF do not intersect. Let CFCF meet Γ1,Γ2\Gamma_1,\Gamma_2 again at K,LK,L respectively, and DEDE meet Γ1,Γ2\Gamma_1,\Gamma_2 at M,NM,N respectively. If the circumcircles of △ALM\triangle ALM and △BKN\triangle BKN are tangent, prove that the radii of these two circles are equal.
geometrycircumcircle
Concyclic with China

Source: CSMO 2019 Grade 11 Problem 2

7/30/2019
ABCDABCD is a parallelogram with ∠BAD≠90\angle BAD \neq 90. Circle centered at AA radius BABA denoted as ω1\omega _1 intersects the extended side of AB,CBAB,CB at points E,FE,F respectively. Suppose the circle centered at DD with radius DADA, denoted as ω2\omega _2, intersects AD,CDAD,CD at points M,NM,N respectively. Suppose EN,FMEN,FM intersects at GG, AGAG intersects MEME at point TT. MFMF intersects ω1\omega _1 at Q≠FQ \neq F, and ENEN intersects ω2\omega _2 at P≠NP \neq N. Prove that G,P,T,QG,P,T,Q concyclic.
geometry