MathDB

3

Part of 2009 Romanian Master of Mathematics

Problems(1)

Prove that the four lines AiOi are concurrent or parallel

Source: Romanian Master in Mathematics 2009, Problem 3

3/7/2009
Given four points A1,A2,A3,A4 A_1, A_2, A_3, A_4 in the plane, no three collinear, such that A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, denote by Oi O_i the circumcenter of △AjAkAl \triangle A_j A_k A_l with \{i,j,k,l\} \equal{} \{1,2,3,4\}. Assuming ∀iAi≠Oi, \forall i A_i \neq O_i , prove that the four lines AiOi A_iO_i are concurrent or parallel.
Nikolai Ivanov Beluhov, Bulgaria
geometrycircumcircle3D geometrytetrahedronprismrhombus