Reference: Engel's Problem Solving Strategies, c.f. 12.2 Problem #1.
Problem. $\Delta ABC$, $\Delta A'B'C$ are regular triangles with the same orientation. Let $P,Q,R$ be the midpoints of $AB',BC,A'C$. Show $\Delta PQR$ is regular.
So I'm fairly certain I have a strategy for this problem. The issue is that my strategy is basically a copycat of a strategy presented in examples preceding the exercise which I don't completely understand. I think I've applied it correctly but I'm not getting how the regularity follows.
Solution. Starting at $R$, apply a dilation from $A'$ with factor 2 and arrive at $C$. Then apply a rotation about $B$ of $60^\circ$ to arrive at $A$. Finally, apply a dilation away from $B'$ of factor $\frac{1}{2}$ to arrive at $P$. Apply the same sequence of transformations starting at $Q$ and notice it is a fixed point (the dilations' lengths cancel each other and the rotation accounts for the angular displacement between the vertices $A',B'$ from which $Q$ stretched). Since the isometry has a fixed point, it is a rotation about $Q$ of $60^\circ$, so $\Delta PQR$ must be regular.
In a preceding example, Engel used this argument in an almost identical situation. He applies transformations around various points in context arriving at an isometry that mapped one vertex of the triangle in question to another one; then, he showed that the third vertex was fixed. The conclusion was regularity of the triangle in question.
Why?
