Coxeter- Exercise 5.3.1
Exercise 5.3.1 Given six collinear points A, B, C, D, E, F, consider the three involutions (AB)(DE), (BC)(EF), (CD)(FA). If any two of these involutions have a common pair, all three have a common pair.
Drag any one point O, O1, or O2 onto another one of those points or stack all three, at least two of the involutions will have a common pair forcing the third to be in common with the other two as well.