MathDB

6

Part of 2013 Oral Moscow Geometry Olympiad

Problems(2)

minimal distance between circumcenters

Source: 2013 Oral Moscow Geometry Olympiad grades 8-9 p6

8/15/2019
Let ABCABC be a triangle. On its sides ABAB and BCBC are fixed points C1C_1 and A1A_1, respectively. Find a point P P on the circumscribed circle of triangle ABCABC such that the distance between the centers of the circumscribed circles of the triangles APC1APC_1 and CPA1CPA_1 is minimal.
geometryMinimalminimumdistanceCircumcentercircumcircle
lines intersect on circle, isosceles trapezoid, incircles related

Source: 2013 Oral Moscow Geometry Olympiad grades 10-11 p6

8/17/2019
The trapezoid ABCDABCD is inscribed in the circle ww (AD//BCAD // BC). The circles inscribed in the triangles ABCABC and ABDABD touch the base of the trapezoid BCBC and ADAD at points PP and QQ respectively. Points XX and YY are the midpoints of the arcs BCBC and ADAD of circle ww that do not contain points AA and BB respectively. Prove that lines XPXP and YQYQ intersect on the circle ww.
geometrytrapezoidincircleconcurrencyconcurrentarc midpoint