MathDB

5

Part of 2004 Oral Moscow Geometry Olympiad

Problems(2)

PK = LQ wanted, 2 equilateral triangles outside convex cyclic ABCD

Source: 2004 Oral Moscow Geometry Olympiad grade 9 p5

10/27/2020
The diagonals of the inscribed quadrilateral ABCDABCD meet at the point MM, ∠AMB=60o\angle AMB = 60^o. Equilateral triangles ADKADK and BCLBCL are built outward on sides ADAD and BCBC. Line KLKL meets the circle circumscribed ariound ABCDABCD at points PP and QQ. Prove that PK=LQPK = LQ.
geometryequal segmentscyclic quadrilateral
cyclic quadr. by projections of point og circumcircle of trapezoid on sides

Source: 2004 Oral Moscow Geometry Olympiad grade 10 p5

10/28/2020
Trapezoid ABCDABCD with bases ABAB and CDCD is inscribed in a circle. Prove that the quadrilateral formed by orthogonal projections of any point of this circle onto lines AC,BC,ADAC, BC, AD and BDBD is inscribed.
geometrycyclic quadrilateraltrapezoid