Inverse Similarity through Reflections

What Might This Be About?

Problem

Let $P$ and $Q$ be two points in $\Delta ABC;$ lines $\ell_a,$ $\ell_b,$ $\ell_c$ through $Q$ be perpendicular to the sides $BC,$ $AC,$ and $AB,$ respectively; $P_a,$ $P_b,$ $P_c$ be reflections of $P$ in $\ell_a,$ $\ell_b,$ $\ell_c.$

Inverse Similarity through Reflections, problem

Then $\Delta P_aP_bP_c$ is inversely similar to $\Delta ABC.$

Proof

The proof is by angle chasing for which the following diagram may serve as a road map.

Inverse Similarity through Reflections, solution

Extension

Define $Q_a,$ $Q_b,$ $Q_c$ as the reflections of $Q$ in $P_a,$ $P_b,$ $P_c.$ Then obviously $\Delta Q_aQ_bQ_c$ is a homothetic image of $\Delta P_aP_bP_c$ and is, therefore similar to the latter. The same is true for $\Delta M_aM_bM_c,$ where points $M$ are midway between the corresponding points $P$ and $Q.$

In particular this is true when $Q=H,$ the orthocenter of $\Delta ABC,$ and $P=O,$ the circumcenter of the latter.

Acknowledgment

The above is a reflection on private communication from Leo Giugiuc.

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