MathDB

6

Part of 2024 Assara - South Russian Girl's MO

Problems(2)

\angle AEX = 65^\circ in regular hexagon

Source: Assara 2024 J6

10/14/2024
In the regular hexagon ABCDEFABCDEF, a point XX was marked on the diagonal ADAD such that ∠AEX=65∘\angle AEX = 65^\circ. What is the degree measure of the angle ∠XCD\angle XCD? A.V.Smirnov, I.A.Efremov
geometry
XY \perp AD Assara

Source: Assara 2024 S6

10/14/2024
The points A,B,C,DA, B, C, D are marked on the straight line in this order. Circle ω1\omega_1 passes through points AA and CC, and the circle ω2\omega_2 passes through points BB and DD. On the circle ω2\omega_2, the point EE is marked so that AB=BEAB = BE, and on the circle ω1\omega_1, the point FF is marked so that CD=CFCD = CF. The line AEAE intersects the circle ω2\omega_2 a second time at point XX, and the line DFDF intersects the circle ω1\omega_1 at point YY. Prove that the XYXY lines and ADAD is perpendicular. A.D.Tereshin
geometry