MathDB

8

Part of 2022 Assara - South Russian Girl's MO

Problems(2)

concurrency in hexagon with AB = BC =CD = DE = EF = FA, AD = BE = CF

Source: 2022 South Russian Girls MO - Assara Juniors p8

2/28/2024
About the convex hexagon ABCDEFABCDEF it is known that AB=BC=CD=DE=EF=FAAB = BC =CD = DE = EF = FA and AD=BE=CFAD = BE = CF. Prove that the diagonals ADAD, BEBE, CFCF intersect at one point.
geometryhexagonconcurrencyconcurrent
AM = AB if (BMC) and (AXY) are tangent

Source: 2022 South Russian Girls MO - Assara Seniors p8

3/1/2024
In parallelogram ABCDABCD, angle AA is acute. Let XX be a point, symmetrical to point CC wrt to straight line ADAD, YY is a point symmetrical to the point CC wrt point DD, and MM is the intersection point of ACAC and BDBD. It turned out, that the circumcircles of triangles BMCBMC and AXYAXY are tangent internally. Prove that AM=ABAM = AB.
geometrytangent circles