Solution to Exercise 2.55.
Perfect numbers like perfect men are very rare..
—Rene Descartes (1596–1650)

Exercise 2.55. For the Poincaré Half-plane, find all lines parallel to the given line through the given point.
      (a)  (2, 1) and  

Note that for a Type II line parallel to the given line, c must be greater than 1 and r must be less than or equal to the distance from c to 1. Hence, the line must satisfy (2 – c)² +1² = r² and   Hence,  Solving yields  Therefore, the line ₂l and the lines  where  and  are parallel to the line ₁l and pass through the point (2, 1).

      (b)  (2, 1) and  

Either the same as part (a) or the line where c < 1 must have r greater than or equal to the distance from c to –1. Hence, the line must satisfy  and  Hence,  Solving the inequality yields  Therefore, the line ₂l and the lines  where  or  and  are parallel to the line  and pass through the point (2, 1).