1102 Constructing tangents to a circle through an external point
Problem:
Given the circle c (with its center O and a circumpoint of it) and a point P outside it. The tangents to c throungh P are to be constructed.
Discussion:
Since Thales' circle theorem cannot be used, such kind of solution is needed that can work both in the Euclidean plane and in the P-model. Therefore first we will consider some properties of absolute geometry that can be useful.
- Each tangent of a circle is perpendicular to the corresponding radius.
- Let two concentric circles be given. Consider any circumpoint on the outer circle and the tangents through that circumpoint to the inner circle. The line segments of the tangents between the inner and outer circumpoints are congruent to each other, and to all other line segments in case a different outer circumpoint is considered.
Construction:
Let T be an arbitrary point of c, construct a tangent to c through T. Now construct a point M on this tangent such that it has the same line segment of tangent as P has. Finally use congruent transformation (e.g. reflection w.r.t. the perpendicular bisector of segment PM) to find one tangent, and then the other one.
Euclid himself felt important that this problem should be solved with as simple means as possible (see Euxlid: Elements III. 17. pp. 115-116). Euclid's construction is indeed somewhat simpler than the one above as he defined point P on line OM. So one can immediately obtain one of the sought tangent points by having the intersection of the given circle and segment OM.
Euclid's construction is remarkable not only because it is its simplicity but also because it can be performed in the P-model and on a sphere as well.