Editing Thales's theorem (section)

==Applications==
===Constructing a tangent to a circle passing through a point===
[[Image:Thales' Theorem Tangents.svg|thumb|Constructing a tangent using Thales's theorem.]]
Thales's theorem can be used to construct the [[tangent]] to a given circle that passes through a given point. In the figure at right, given circle {{mvar|k}} with centre {{mvar|O}} and the point {{mvar|P}} outside {{mvar|k}}, bisect {{mvar|{{overline|OP}}}} at {{mvar|H}} and draw the circle of radius {{mvar|{{overline|OH}}}} with centre {{mvar|H}}. {{mvar|{{overline|OP}}}} is a diameter of this circle, so the triangles connecting OP to the points {{mvar|T}} and {{mvar|T&prime;}} where the circles intersect are both right triangles.

[[File:Root_construction_geometric_mean5.svg|thumb|Geometric method to find <math>\sqrt{p}</math> using the [[geometric mean theorem]] <math>h=\sqrt{pq}</math> with <math>q=1</math>]]

===Finding the centre of a circle===
Thales's theorem can also be used to find the centre of a circle using an object with a right angle, such as a [[set square]] or rectangular sheet of paper larger than the circle.<ref>[https://books.google.com/books?id=9jASBwAAQBAJ&pg=PA183 Resources for Teaching Mathematics: 14–16] Colin Foster</ref> The angle is placed anywhere on its circumference (figure 1). The intersections of the two sides with the circumference define a diameter (figure 2). Repeating this with a different set of intersections yields another diameter (figure 3). The centre is at the intersection of the diameters.
[[File:Thales_theorem_find_circle_centre.svg|thumb|300px|none|Illustration of the use of Thales's theorem and a right angle to find the centre of a circle]]