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3

Part of 2017 Ukrainian Geometry Olympiad

Problems(2)

fixed circle, starting with two intersecting circles and an orthocenter

Source: Ukrainian Geometry Olympiad 2017, IX p3, X p2

12/12/2018
Circles w1,w2{w}_{1},{w}_{2} intersect at points A1{{A}_{1}} and A2{{A}_{2}} . Let BB be an arbitrary point on the circle w1{{w}_{1}}, and line BA2B{{A}_{2}} intersects circle w2{{w}_{2}} at point CC. Let HH be the orthocenter of ΔBA1C\Delta B{{A}_{1}}C. Prove that for arbitrary choice of point BB, the point HH lies on a certain fixed circle.
geometryfixedintersectcirclesLocusLocus problems
AP is tangent to the circumcircle of BLP, starting with a right triangle

Source: Ukrainian Geometry Olympiad 2017 X p3

12/12/2018
On the hypotenuse ABAB of a right triangle ABCABC, we denote a point KK such that BK=BCBK = BC. Let PP be a point on the perpendicular from the point KK to line CKCK, equidistant from the points KK and BB. Let LL be the midpoint of CKCK. Prove that line APAP is tangent to the circumcircle of ΔBLP\Delta BLP.
geometrycircumcircletangentright triangle