91Ó°ÊÓ

Join the vertices \(A, B, C, D\) of a parallelogram to the midpoints of the respective sides \(B C, C D, D A, A B\) so as to form a smaller parallelogram in the middle. Its area is one-fifth that of \(A B C D\). Another such parallelogram is obtained by joining \(A, B, C, D\) to the midpoints of \(C D, D A, A B, B C\). The common part of these two small parallelograms is a centrally symmetrical octagon whose area is one-sixth that of \(A B C D\) [Dörrie 1, p, 40].

From any point \(A_{1}\) on the side \(B C\) of a triangle \(A B C\), draw \(A_{1} B_{1}\) parallel to \(B A\) to meet \(C A\) in \(B_{1}\), then \(B_{1} C_{1}\) parallel to \(C B\) to meet \(A B\) in \(C_{1}\). and then \(C_{1} A_{2}\) parallel to \(A C\) to meet \(B C\) in \(A_{2}\). If \(A_{1}\) is the midpoint of \(B C_{1} A_{2}\) coincides with it. If not, continue the process, drawing \(A_{2} B_{2}\) parallel to \(B A_{2} B_{2} C_{2}\) parallel to \(C B\), and \(C_{2} A_{3}\) parallel to \(A C\). The path is now closed: \(A_{3}\) coincides with \(A_{1}\). (This is called Thomsen's figure. See Geometrical Magic, by Nev R. Mind, Seripta Mathematica, 19 \((1953)\), pp. \(198-200 .)\)

Describe the transformations (i) \(x^{\prime}=x+1 . y^{\prime}=y\); (ii) \(x^{\prime}=a x, y=a y\); (iii) \(x^{\prime}=x+b y, y^{\prime}=y\). (iv) \(x^{\prime}=a x, y^{\prime}=y\).

The product of any even number of affine reflections is an equiaffinity.

In an affinely regular polygon \(P_{0} P_{1} P_{2} \ldots\), the lines \(P_{i} P_{j}\) and \(P_{h} P_{k}\) are parallel whenever \(i+j=h+k\).

What triangles and quadrangles are affinely regular?