On 3N points in a plane

In this note I prove the following theorem: Theorem 1. Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad, the N triangles will all have a common point. The proof depends on three lemmas; the first two are well known, and are quoted without proof; the third I believe to be new, and leads directly to the proof of the theorem.