MathDB

10 concurrent lines

Source: Brazil National Olympiad 2019 #6

November 14, 2019

geometry

Problem Statement

Let

A_1A_2A_3A_4A_5

be a convex, cyclic pentagon with

\angle A_i + \angle A_{i+1} >180^{\circ}

for all

i \in \{1,2,3,4,5\}

(all indices modulo

5

in the problem). Define

B_i

as the intersection of lines

A_{i-1}A_i

and

A_{i+1}A_{i+2}

, forming a star. The circumcircles of triangles

A_{i-1}B_{i-1}A_i

and

A_iB_iA_{i+1}

meet again at

C_i \neq A_i

, and the circumcircles of triangles

B_{i-1}A_iB_i

and

B_iA_{i+1}B_{i+1}

meet again at

D_i \neq B_i

. Prove that the ten lines

A_iC_i, B_iD_i

,

i \in \{1,2,3,4,5\}

, have a common point.

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