Problem

Source: Russian TST 2014, Day 9 P1 (Group NG), P2 (Groups A & B)

Tags: geometry



Finitely many lines are given, which pass through some point $P{}.$ Prove that these lines can be coloured red and blue and one can find a point $Q\neq P$ such that the sum of the distances from $Q{}$ to the red lines is equal to the sum of the distance from $Q{}$ to the blue lines.