AI and BI'. Construct the common perpendicular to II' and AB, which can be done using the ultraparallel theorem because II' and AB are ultraparallel to each other. Name this line CC'. C now ends up being the midpoint of AB. Definitions for complex constructions For the purposes of the following definitions, the following assumptions will be made, which usually cannot be made in hyperbolic geometry Three distinct points create a unique circle Given any two lines, they meet at a unique point (normally, this would contradict the parallel axiom of hyperbolic geometry, since there can be many different lines parallel to the same line) Angle measures have signs. Here, they will be defined in the following way: Consider a triangle XYZ. The sign of angle ᗉ XYZ is positive if and only if the direction of the path along the shortest arc from side XY to side YZ is counterclockwise. The picture of the triangle on the right describes this. To make a comparison, when working with the unit circle, the angle measure is positive when going counterclockwise and negative when going clockwise. Cyclic quadrilaterals A quadrilateral is cyclic if the two opposite vertices add up to pi radians or 180 degrees. Also, if a quadrilateral is inscribed in a circle in a way that all of its vertexes lie on the circle, it is cyclic. Pseudoaltitudes Consider triangle ABC where the points are labeled in a clockwise manner so all angles are positive. Let X be a point moving along BC from B to C. As X moves closer to C, angle ᗉAXB will decrease and angle ᗉ AXC will increase. When X is close enough to B, ᗉ AXB > ᗉ AXC. When X is close enough to C, ᗉ AXB < ᗉ AXC. This means that at some point, X will be in a position where ᗉ AXB = ᗉ AXC. When X is in this position, it is defined as the foot of the pseudoaltitude from vertex A. The pseudoaltitude would then be the line segment AX. Pseudolengths Let dE(A,B) denote the pseudolength for a given hyperbolic line segment AB. Let a transformation move A to the center of a Poincaré disk with a radius equalling 1. The pseudolength dE(A,B) is the length of this segment in Euclidean geometry. Homothety Given a point P, a point A where A is the center of the homothety,