SHARYGIN 2014 Problem 23

Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic. Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that a) $A, B, C_1, D_1$ are concyclic; b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.