MathDB

3

Part of 2022 Czech-Polish-Slovak Junior Match

Problems(2)

convex pentagon ABCD fits in a circle with radius 2/3 AD

Source: 2022 Czech-Polish-Slovak Match Junior, individual p3 CPSJ

9/3/2022
Given is a convex pentagon ABCDEABCDE in which ∠A=60o\angle A = 60^o, ∠B=100o\angle B = 100^o, ∠C=140o\angle C = 140^o. Show that this pentagon can be placed in a circle with a radius of 23AD\frac23 AD.
geometrypentagon
PQ//BC wanted, FB = BD, DC = CE

Source: 2022 Czech-Polish-Slovak Match Junior, team p3 CPSJ

9/3/2022
The points D,E,FD, E, F lie respectively on the sides BCBC, CACA, ABAB of the triangle ABC such that FB=BDF B = BD, DC=CEDC = CE, and the lines EFEF and BCBC are parallel. Tangent to the circumscribed circle of triangle DEFDEF at point FF intersects line ADAD at point PP. Perpendicular bisector of segment EFEF intersects the segment ACAC at QQ. Prove that the lines PQP Q and BCBC are parallel.
geometryparallelequal segments