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## Basics: Geometrical elements

This sample is composition of straight lines, circles, circular arcs, and text. The layout of the entire (main) figure is defined by the three points of the triangle ABC. The position of the inset figure would also need to be defined. The line styles, and drawn annotations would be specified. Ideally the locations of the text elements would be defined in term of the lines and corners to which they are adjacent. For instance, the label `C' might be placed in the corner of the lines C'' to C and C to A.

The construction of the figure from the triangle ABC is quite simple. The line BC is extended so that the length A to B' equals A to C, and B to B'' equals A to B. The points C' and C'' are similarly located. The incircles and excircles are constructed, and the line through E and F is tangential to the excircle and parallel to BC. The points E, F, E' and F' are found from intersections. The angle CBD is , and is its complementary angle. All other angles (and parallel-line relationships) follow by geometrical principles. Proving that the lines C'' to B'' and B'' to E' are perpendicular is the most difficult step. One means is to use the inset figure. This is the quadrilateral BDCG with a circle constructed on diameter DG. A copy is made of the quadrilateral, rotated through 180 degrees.

Drawing this figure in XFig required painstaking effort. The whole design is based on Pythagorean triples, in order that every labelled point except for F be on a grid point. The line to F and some of the inset elements are constructed using scaling and rotation. Once Adage is developed, it will be fun to redraw the figure and experiment by moving point A around relative to B and C.

The relative sizes of the incircle and excircle. Starting with the triangle ABC construct the incircle and an excircle. The rest of the drawing illustrates interrelationships between lines and points of intersection. Creation method: XFig, ps output.

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