The figure below is the stereographic image of the equator e (thick line circle) and two orthogonal great circles c1 and c2. For your convenience the Euclidean centers E1 and E2 and S of c1 and c2 and e are also shown. The exercise in this problem is to construct a number of points with an unmarked straightedge alone (and the figure itself). No compass is needed and no measuring device.
· Draw points A and B in the Euclidean plane. Draw any point P on line AB, not between A and B. Construct a circle c with center P so that the inversion of A in c is B.
· Draw a circle D and points A, B, C, D on the circle. Construct a circle c that inverts A to B and C to D. Can you construct such a circle if A, B, C, D are not all on the same circle?
· Fill in the table that was handed out Wednesday.
The figure below is a stereographic image of 3 great circles which are extended sides of a spherical triangle ABC.
· How many spherical triangles are defined by the 3 great circles?
· Are some of them congruent? Which ones? Mark congruent triangles and angles equal to vertex angles a, b, c of triangle ABC.
· Let T be the area of triangle ABC. Give names to the other areas of the other triangles. Write down as many area relations as you can, using the area formula for lunes.
· Can you figure out T from the angles a, b, c?
