MathDB

1

Part of 2014 Oral Moscow Geometry Olympiad

Problems(2)

circumcenter coincides with excenter , <A=45^o

Source: 2014 Oral Moscow Geometry Olympiad grades 8-9 p1

8/9/2019
In triangle ABC,∠A=45o,BHABC, \angle A= 45^o, BH is the altitude, the point KK lies on the ACAC side, and BC=CKBC = CK. Prove that the center of the circumscribed circle of triangle ABKABK coincides with the center of an excircle of triangle BCHBCH.
geometryCircumcenterexcenter
concurrency in a trapezoid, midpoints, height related

Source: 2014 Oral Moscow Geometry Olympiad grades 10-11 p1

8/9/2019
In trapezoid ABCDABCD: BC<AD,AB=CD,KBC <AD, AB = CD, K is midpoint of AD,MAD, M is midpoint of CD,CHCD, CH is height. Prove that lines AM,CKAM, CK and BHBH intersect at one point.
geometrytrapezoidconcurrencyconcurrent