As viewed, the perpendicular bisectors of the sides AB and BC of the non E triangle ABC meet at the point P. The third side's perpendicular bisector would meet the other two at P as well, as in Euclidean geometry. This point is the centre of the circum-circle,a nonE circle that passes through the three vertices of the triangle. But the matter is not quite as clear cut as that. Move the point B towards the opposite side AC. The centre of the circle P moves towards the disc horizon until it vanishes when the two bisectors are parallel. At this point on the disc it is the horocycle that passes through the three vertices. As the bisectors become ultra parallel the construction is as follows. Draw the hypercycle (the equi-distance line) through B to the unique perpendicular bisector of the two ultra parallel lines. It is this curve , shown in red, that passes through all three vertices. |