Problem 1
Six points are chosen on the sides of an equilateral triangle : on , on and on , such that they are the vertices of a convex hexagon with equal side lengths. Prove that the lines , and are concurrent.
Step 1 of 6: Close up the hexagon
In plain words
Walking all the way around a closed hexagon and back to the start means the six edge vectors must cancel out.
Detailed analysis
The six points, taken in this cyclic order, form a closed hexagon, so the sum of its six directed edges is the zero vector.