MathDB

4

Part of 2010 Sharygin Geometry Olympiad

Problems(4)

A1B1C1D1 is inscribed in a circle centered at N (4)

Source:

10/28/2010
The diagonals of a cyclic quadrilateral ABCDABCD meet in a point N.N. The circumcircles of triangles ANBANB and CNDCND intersect the sidelines BCBC and ADAD for the second time in points A1,B1,C1,D1.A_1,B_1,C_1,D_1. Prove that the quadrilateral A1B1C1D1A_1B_1C_1D_1 is inscribed in a circle centered at N.N.
geometrycircumcirclecyclic quadrilateralgeometry proposed
common tangent to unequal circles inscribed in equal consecutive angles

Source: Sharygin 2010 Final 8.4

10/2/2018
Circles ω1\omega_1 and ω2\omega_2 inscribed into equal angles X1OYX_1OY and YOX2Y OX_2 touch lines OX1OX_1 and OX2OX_2 at points A1A_1 and A2A_2 respectively. Also they touch OYOY at points B1B_1 and B2B_2. Let C1C_1 be the second common point of A1B2A_1B_2 and ω1,C2\omega_1, C_2 be the second common point of A2B1A_2B_1 and ω2\omega_2. Prove that C1C2C_1C_2 is the common tangent of two circles.
geometrytangentcirclesequal anglescommon tangents
triangle construction related to touchpoints of incircle with triangle sides

Source: Sharygin 2010 Final 9.4

10/6/2018
In triangle ABCABC, touching points A′,B′A', B' of the incircle with BC,ACBC, AC and common point GG of segments AA′AA' and BB′BB' were marked. After this the triangle was erased. Restore it by the ruler and the compass.
geometryconstructionincircle
parallelogram criterion related to projections on 2 concentric circles

Source: Sharygin 2010 Final 10.4

11/25/2018
Projections of two points to the sidelines of a quadrilateral lie on two concentric circles (projections of each point form a cyclic quadrilateral and the radii of circles are different). Prove that this quadrilateral is a parallelogram.
geometryparallelogramconcentric circlesconcentricprojectionsinscribed