O is the unique fixed point of f. Thus, given a quadrangle EFGH, there is a unique other quadrangle IJKO, whose medial lines carry the sides of the given quadrangle in a given order. Here we reflect on the sides e, f, g, h on that order. There are though 24 other possible permutations of these letters, resulting to 24 quadrangles with the stated property. Are they different all of them?

How can we find quickly O? The two questions are related. See the answer below.

Denote by (efgh) the order of the reflections. The partial compositions (ef) and (gh) are rotations, hence they can represented by some other reflections (e'f') and (f'h'). For this the angles <(e'f') and <(f'h') must be equal to the corresponding <(ef) and <(gh). From this remark follows the construction of O as the intersection point of e' and h'.

Consider now the order (fehg) and do the corresponding construction. This gives the triangle EGH on the diagonal EG, which is symmetric to the EGO w.r. to that diagonal. J is the corresponding rotation-center and if we do the reflexions in the order (fehg) we get the same quadrangle IJKO in reverse orientation.

In total we see that the reflections in the orders (efgh) and its 4 circular permutations and (fehg) and its 4 circular permutations, define all of them rotation-centers that are vertices of the quadrangle IJKO. Thus 8 out of the 24 quadrangles coincide.

An analogous argument shows that only 3 quandrangles exist, such that their middle-lines are the sides of a given quadrangle.

For the final picture look at the file: ReflectingOnPolygonSides2.html .

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