Application
Thales' theorem can be used to construct the tangent to a given circle that passes through a given point. (See figure.) Given a circle k, with a center O, and a point P outside of the circle, we want to construct the (red) tangent(s) to k that pass through P. Suppose the (as yet unknown) tangent t touches the circle in the point T. From symmetry, it is clear that the radius OT is orthogonal to the tangent. So construct the midpoint H between O and P, and draw a circle centered at H through O and P. By Thales' theorem, the sought point T is the intersection of this circle with the given circle k, because that is the point on k that completes a right triangle OTP.
Since the two circles intersect at two points, we can construct both tangents in this fashion.
Read more about this topic: Thales' Theorem
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