MathDB

5

Part of 2023 Yasinsky Geometry Olympiad

Problems(5)

one arc is equal to the sum of the other two, in circle with diameter AT

Source: 2023 Yasinsky Geometry Olympiad VIII-IX basic p5 , Ukraine

1/21/2024
Point OO is the center of the circumscribed circle of triangle ABCABC. Ray AOAO intersects the side BCBC at point TT. With ATAT as a diameter, a circle is constructed. At the intersection with the sides of the triangle ABCABC, three arcs were formed outside it. Prove that the larger of these arcs is equal to the sum of the other two.
(Oleksii Karliuchenko)
geometryarcs
P lies on BC iff P is midpoint of BC

Source: V.A. Yasinsky Geometry Olympiad 2023 VIII p5 , Ukraine

12/12/2023
Let ABCABC be a triangle and ℓ\ell be a line parallel to BCBC that passes through vertex AA. Draw two circles congruent to the circle inscribed in triangle ABCABC and tangent to line ℓ\ell, ABAB and BCBC (see picture). Lines DEDE and FGFG intersect at point PP. Prove that PP lies on BCBC if and only if PP is the midpoint of BCBC.
(Mykhailo Plotnikov)
https://cdn.artofproblemsolving.com/attachments/8/b/2dacf9a6d94a490511a2dc06fbd36f79f25eec.png
midpointgeometry
angle bisector АW tangent to (CNW)

Source: 2023 Yasinsky Geometry Olympiad X-XI basic p5 , Ukraine

1/21/2024
The extension of the bisector of angle AA of triangle ABCABC intersects with the circumscribed circle of this triangle at point WW. A straight line is drawn through WW, which is parallel to side ABAB and intersects sides BCBC and ACAC , at points NN and KK, respectively. Prove that the line AWAW is tangent to the circumscribed circle of △CNW\vartriangle CNW.
(Sergey Yakovlev)
geometrytangentangle bisector
AD _|_ BC if KX _|_ AB and KY_|_ AC

Source: V.A. Yasinsky Geometry Olympiad 2023 IX p5 , Ukraine

12/12/2023
Let II be the center of the circle inscribed in triangle ABCABC. The inscribed circle is tangent to side BCBC at point KK. Let XX and YY be points on segments BIBI and CICI respectively, such that KX⊥ABKX \perp AB and KY⊥ACKY\perp AC. The circumscribed circle around triangle XYKXYK intersects line BCBC at point DD. Prove that AD⊥BCAD \perp BC.
(Matthew Kurskyi)
geometryperpendicular bisector
circumcenter construction given incenter and touchpoints of (I) with sides

Source: V.A. Yasinsky Geometry Olympiad 2023 X-XI p5 , Ukraine

12/13/2023
Let ABCABC be a scalene triangle. Given the center II of the inscribe circle and the points K1K_1, K2K_2 and K3K_3 where the inscribed circle is tangent to the sides BCBC, ACAC and ABAB. Using only a ruler, construct the center of the circumscribed circle of triangle ABCABC.
(Hryhorii Filippovskyi)
geometryCircumcenterconstruction