MathDB

10.9

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(2)

comp. geo with cyclic quad, related to Newton−Gauss line

Source: II Soros Olympiad 1995-96 R1 10.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/3/2024
The opposite sides of a quadrilateral inscribed in a circle intersect at points KK and LL. Let FF be the midpoint of KLKL, EE and GG be the midpoints of the diagonals of the given quadrilateral. It is known that FE=aFE = a, FG=bFG = b. Calculate KLKL in terms of aa and b.b.
(It is known that the points FF, EE and GG lie on the same straight line. This is true for any quadrilateral, not necessarily inscribed. The indicated straight line is sometimes called the Newton−Gauss line. This fact can be used without proof in proving the problem, as it is known).
geometryNewton linecyclic quadrilateral
computational with inscribed trapezoid

Source: II Soros Olympiad 1995-96 R3 10.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/6/2024
Trapezoid ABCDABCD with bases ADAD and BCBC is inscribed in a circle, MM is the intersection of of its diagonals. A straight line passing through MM perpendicular to the bases intersects BCBC at pointK K, and the circle at point LL, where LL is the one of the two intersection points for which MM lies on the segment KLKL. It is known that MK=aMK = a, LM=bLM = b. Find the radius of the circle tangent to the segments AMAM, BMBM and the circle circumscribed around ABCDABCD.
geometrytrapezoid