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Part of 2020 Ukrainian Geometry Olympiad - April

Problems(4)

<BCP < <BHP wanted, parallelogram related to orthocenter

Source: Ukrainian Geometry Olympiad 2020, IX p3, X p2

4/27/2020
Let HH be the orthocenter of the acute-angled triangle ABCABC. Inside the segment BCBC arbitrary point DD is selected. Let PP be such that ADPHADPH is a parallelogram. Prove that ∠BCP<∠BHP\angle BCP< \angle BHP.
geometryparallelogramorthocenteranglesangle inequalities
// wanted, circumcircle, equal segments, perp (Ukr. Geom. Olympiad '20 VIII p3)

Source:

6/8/2020
Triangle ABCABC. Let B1B_1 and C1C_1 be such points, that AB=BB1,AC=CC1AB= BB_1, AC=CC_1 and B1,C1B_1, C_1 lie on the circumscribed circle Γ\Gamma of △ABC\vartriangle ABC. Perpendiculars drawn from from points B1B_1 and C1C_1 on the lines ABAB and ACAC intersect Γ\Gamma at points B2B_2 and C2C_2 respectively, these points lie on smaller arcs ABAB and ACAC of circle Γ\Gamma respectively, Prove that BB2∥CC2BB_2 \parallel CC_2.
geometrycircumcircleparallelperpendicularequal segments
equal angles wanted, two intersecting circles both tangent to given rays

Source: Ukrainian Geometry Olympiad 2020, XI p3

4/27/2020
The angle POQPOQ is given (OPOP and OQOQ are rays). Let MM and NN be points inside the angle POQPOQ such that ∠POM=∠QON\angle POM = \angle QON and ∠POM<∠PON\angle POM < \angle PON. Consider two circles: one touches the rays OPOP and ONON, the other touches the rays OMOM and OQOQ. Denote by BB and CC the points of their intersection. Prove that ∠POC=∠QOB\angle POC = \angle QOB.
geometryequal anglescirclesTangents
midpoint M lies on line X_1X_2,also lies on Y_1Y_2 , interecting circles

Source: Ukrainian Geometry Olympiad 2020, X p3

4/27/2020
The circles ω1\omega_1 and ω2\omega_2 intersect at points AA and BB, point MM is the midpoint of ABAB. On line ABAB select points S1S_1 and S2S_2. Let S1X1S_1X_1 and S1Y1S_1Y_1 be tangents drawn from S1S_1 to circle ω1\omega_1, similarly S2X2S_2X_2 and S2Y2S_2Y_2 are tangents drawn from S2S_2 to circles ω2\omega_2. Prove that if the point MM lies on the line X1X2X_1X_2, then it also lies on the line Y1Y2Y_1Y_2.
geometrycirclesTangentsmidpoint