Reim's Similar Coins III

What Might This Be About?

Problem

Let two circles cross at points $A$ and $B;$ $E,F,E'F'$ are four points on one of the circles, $EA$ and $E'B$ meet the other circle at $H$ and $H',$ respectively; $FB,F'A$ meet it at $G$ and $G',$ respectively. Assume $EF\parallel E'F'.$

Reim's Similar Coins I - problem

Then $GH\parallel EF\parallel E'F'\parallel G'H'.$

Solution

The problem can be solved directly, as suggested by the diagram below.

Reim's Similar  Coins III - solution

But, in fact, it is a direct consequence of Reim's Similar Coins theorem.

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