1111 Constructing lines that are perpendicular to a given line and tangent to a given circle
Problem:
In the hyperbolic plane given circle s (with center P and a circumpoint), and line a. Construct the lines perpendicular to a and tangential to s.
Discussion:
In Euclidean geometry this is not a real challenge — there is only one line through P parallel to a. But in our case the problem is much more complex, so it is worth applying George Pólya's (1887-1985) method, namely, we assume what is required to be done as already done. Draw tangent ea to the given circle s through an arbitrary point EA, consider an arbitrary point A on it, and the line through A and perpendicular to ea. This line will be used as an input in another problem later. Let T be the perpendicular projection of center P of circle s on line a.
By recalling Bolyai's construction (Fig. 4) we recognize a Lambert quadrilateral in the construction, with three right angles in quadrilateral PEAAT.
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Solution:
In the present case, firstly we construct the perpendicular projection T of center P on a. Next, the intersection point A0 of circle s and ray [P,V1) asymptotically parallel to a are constructed (Fig. 6b). The orthogonal projection of point A0 to line (PT) yields point T0. At this stage we have two options to continue:
- Point EA is obtained by intersecting circle s and ray [PV0) — here point V0 is a point of line (A0T0) in infinity. In this case we must show that line eA through EA and tangential to s is perpendicular to a.
- Let A be the mirror image of point A0 with respect to the perpendicular bisector of segment T0T, and line eA the perpendicular line through A to a. In this case we must show that eA is tangential to circle s, that is, the perpendicular projection of P on eA goes through the circle s.