MathDB

5

Part of 2020 Ukrainian Geometry Olympiad - December

Problems(4)

angle wanted, <ACB=75^o, altitudes, parallels, circumcircle

Source: December 2020 Ukraine Geometry Olympiad XI p5

12/20/2020
In an acute triangle ABCABC with an angle ∠ACB=75o\angle ACB =75^o, altitudes AA3,BB3AA_3,BB_3 intersect the circumscribed circle at points A1,B1A_1,B_1 respectively. On the lines BCBC and CACA select points A2A_2 and B2B_2 respectively suchthat the line B1B2B_1B_2 is parallel to the line BCBC and the line A1A2A_1A_2 is parallel to the line ACAC . Let MM be the midpoint of the segment A2B2A_2B_2. Find in degrees the measure of the angle ∠B3MA3\angle B_3MA_3.
geometryanglesaltitudescircumcircle
computational , OG // BC, <ACB=45^o, centroid, circumcenter

Source: December 2020 Ukraine Geometry Olympiad IX p5, X p4

12/20/2020
Let ABCABC be an acute triangle with ∠ACB=45o\angle ACB = 45^o, GG is the point of intersection of the medians, and OO is the center of the circumscribed circle. If OG=1OG =1 and OG∥BCOG \parallel BC, find the length of BCBC.
geometryparallelCentroidCircumcentercircumcircle
computational, starting with intersecting circles

Source: December 2020 Ukraine Geometry Olympiad X p5 , XI p4

12/20/2020
Let Γ1\Gamma_1, Γ2\Gamma_2 be two circles, whereΓ1 \Gamma_1 has a smaller radius, intersect at two points AA and BB. Points C,DC, D lie on Γ1\Gamma_1, Γ2\Gamma_2 respectively so that the point AA is the midpoint of the segment CDCD . LineCB CB intersects the circle Γ2\Gamma_2 for the second time at the point FF, line DBDB intersects the circle Γ1\Gamma_1 for the second time at the point EE. The perpendicular bisectors of the segments CDCD and EFEF intersect at a point PP. Knowing that CA=12CA =12 and PE=5PE = 5 , find APAP.
geometrycircles
computational with circumcenter, AB =1, AO = AC = 2, OD = OE, BD =\sqrt2 EC

Source: December 2020 Ukraine Geometry Olympiad VIII p5 , IX p4

12/21/2020
Let OO is the center of the circumcircle of the triangle ABCABC. We know that AB=1AB =1 and AO=AC=2AO = AC = 2 . Points DD and EE lie on extensions of sides ABAB and ACAC beyond points BB and CC respectively such that OD=OEOD = OE and BD=2ECBD =\sqrt2 EC. Find OD2OD^2.
geometryCircumcenterequal segments