MathDB
Lengths of arcs

Source: 2019 Second Round - Poland

July 8, 2019
geometryarcscyclic quadrilateral

Problem Statement

A cyclic quadrilateral ABCDABCD is given. Point K1,K2K_1, K_2 lie on the segment ABAB, points L1,L2L_1, L_2 on the segment BCBC, points M1,M2M_1, M_2 on the segment CDCD and points N1,N2N_1, N_2 on the segment DADA. Moreover, points K1,K2,L1,L2,M1,M2,N1,N2K_1, K_2, L_1, L_2, M_1, M_2, N_1, N_2 lie on a circle ω\omega in that order. Denote by a,b,c,da, b, c, d the lengths of the arcs N2K1,K2L1,L2M1,M2N1N_2K_1, K_2L_1, L_2M_1, M _2N_1 of the circle ω\omega not containing points K2,L2,M2,N2K_2, L_2, M_2, N_2, respectively. Prove that \begin{align*} a+c=b+d. \end{align*}