MathDB

2

Part of 2015 Korea National Olympiad

Problems(2)

Easy Geometry

Source: 2015 Korean Mathematical Olympiad P2

11/1/2015
Let the circumcircle of △ABC\triangle ABC be ω\omega. A point DD lies on segment BCBC, and EE lies on segment ADAD. Let ray AD∩ω=FAD \cap \omega = F. A point MM, which lies on ω\omega, bisects AFAF and it is on the other side of CC with respect to AFAF. Ray ME∩ω=GME \cap \omega = G, ray GD∩ω=HGD \cap \omega = H, and MH∩AD=KMH \cap AD = K. Prove that B,E,C,KB, E, C, K are cyclic.
geometrycircumcircle
Angles in a isosceles trapezoid

Source: 2015 Korean Mathematical Olympiad P6

11/1/2015
An isosceles trapezoid ABCDABCD, inscribed in ω\omega, satisfies AB=CD,AD<BC,AD<CDAB=CD, AD<BC, AD<CD. A circle with center DD and passing AA hits BD,CD,ωBD, CD, \omega at E,F,P(≠A)E, F, P(\not= A), respectively. Let AP∩EF=QAP \cap EF = Q, and ω\omega meet CQCQ and the circumcircle of △BEQ\triangle BEQ at R(≠C),S(≠B)R(\not= C), S(\not= B), respectively. Prove that ∠BER=∠FSC\angle BER= \angle FSC.
geometrytrapezoidcircumcircle