MathDB

3

Part of 2021 Bundeswettbewerb Mathematik

Problems(2)

Angle between points of intersections of angular bisectors with opposite sides

Source: Bundeswettbewerb Mathematik 2021, Round 1 - Problem 3

4/12/2021
Consider a triangle ABCABC with ∠ACB=120∘\angle ACB=120^\circ. Let A’,B’,C’A’, B’, C’ be the points of intersection of the angular bisector through AA, BB and CC with the opposite side, respectively. Determine ∠A’C’B’\angle A’C’B’.
geometrygeometry proposed
A classical geometry problem

Source: Bundeswettbewerb Mathematik 2021, Round 2 - Problem 3

9/22/2021
We are given a circle kk and a point AA outside of kk. Next we draw three lines through AA: one secant intersecting the circle kk at points BB and CC, and two tangents touching the circlekk at points DD and EE. Let FF be the midpoint of DEDE.
Show that the line DEDE bisects the angle ∠BFC\angle BFC.
geometrygeometry proposedtangentangle bisector