 ### Challenge from 8. December

Straight harmony

Author: Luise Fehlinger

Project: ZE-AP1 – Teachers at University

Challenge:

The leadlight window in Santa’s workshop was destroyed. The window had the shape of a trapezium and was divided into two parts by a beam at its midsegment. The upper part was coloured red and the lower green (see Figure 1). Of course, the gnomes are eager to repair the broken window at once. Figure 1: The window has the form of a trapezium with bases a and b, and legs b and d. The trapezium ABCD is divided into two smaller trapezia by the parallel m.

However, Santa is quite stressed and thus requests a new partition that is more harmonic than the one before. He believes that, if one has to build a new window anyway, one can consider a new design which is much more pleasant:

The gnomes are to build a new beam m parallel to the bases such that the length of this beam m equals the harmonic mean of the bases a and c. The harmonic mean is defined to be the inverse of the arithmetic mean of the inverses. That is, the harmonic mean of a and c is given by Now, the gnomes are discussing how to realise Santa’s demand.

Which of the following constructions does the trick?

Artwork: Sonja Rörig 1. Marek says: Santa should not make such a fuss about the new window. The midline already has the required length.

2. Nadia suggests the following construction: Draw a circle of radius c around A. Denote the intersection of this circle with AB by E. Construct the perpendicular bisector of AE. Denote an intersection of this bisector with the circle around B through A by F. Then, AF has the required length.

3. Ida is quite sure that the parallel beam has to be installed such that the areas of the red and green window are the same.

4. Jonas divides the height of the trapezium at the ratio of c (below) to a (above). There, he wants to install the parallel beam.

5. Hannah suggests to build the beam parallel to the bases through the barycentre of the trapezium.

6. Employing shearings, Rasmus constructs two rectangles from a square with edge length 1: one with edge lengths a and 1a, and one with edge lengths c and 1c. Afterwards, he consecutively marks the lengths 1a and 1c on a half-line, and bisects the resulting line segment. The thus constructed line segment has the requested length.

7. Lina wants to keep it short and simple: She wants to use the line parallel to the bases that runs through the intersection of the diagonals of the trapezium as the beam that separates the two parts of the window.

8. Jolanda draws the perpendicular through D onto the base a; then, the diagonal AC. Through their intersection, she constructs the parallel to the bases.

9. Cornelius constructs the perpendicular bisectors of the legs b and d. Then, he draws the parallel to the bases through their intersection.

10. Milena is positive that Santa just wants to play tricks on the gnomes and that it is impossible to construct such a beam.