It might be said, for example, that in planning a Gothic window this drawing is needed. The arc _BC_ is drawn with _A_ as a center and _AB_ as a radius. The small arches are described with _A_, _D_, and _B_ as centers and _AD_ as a radius. The center _P_ is found by taking _A_ and _B_ as centers and _AE_ as a radius. How may the points _D_, _E_, and _F_ be found? Draw the figure. From the study of the rectilinear figures suggested by such a simple pattern the properties of the equilateral triangle may be inferred. The Gothic window also offers some interesting possibilities in connection with the study of the square. For example, the illustration given on page 223 shows a number of traceries involving the construction of a square, the bisecting of angles, and the describing of circles.[72] [Illustration] The properties of the square, a figure now easily constructed by the pupils, are not numerous. What few there are may be brought out through the study of art forms, if desired. In case these forms are shown to a class, it is important that they should be selected from good designs. We have enough poor art in the world, so that geometry should not contribute any more. This illustration is a type of the best medieval Gothic parquetry.[73] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES] Even simple designs of a semipuzzling nature have their advantage in this connection. In the following example the inner square contains all of the triangles, the letters showing where they may be fitted.[74] Still more elaborate designs, based chiefly upon the square and circle, are shown in the window traceries on page 225, and others will be given in connection with the study of the regular polygons. [Illustration] Designs like the figure below are typical of the simple forms, based on the square and circle, that pupils may profitably incorporate in any work in art design that they may be doing at the time they are studying the circle and the problems relating to perpendiculars and squares. [Illustration] Among the applications of the problem to draw a tangent to a given circle is the case of the common tangents to two given circles. Some authors give this as a basal problem, although it is more commonly given as an exercise or a corollary. One of the most obvious applications of the idea is that relating to the transmission of circular motion by means of a band over two wheels,[75] _A_ and _B_, as shown on page 226. [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]