Problem
GEO-B1-M05-P020 Tangents From One Point
#20
★★★☆☆ Level 3 of 5
From point \(T\), tangents \(TA\) and \(TB\) are drawn to a circle with centre \(O\). Prove that \(TA=TB\), and that \(OT\) bisects angle \(ATB\).
Compare right triangles \(OTA\) and \(OTB\).
Since \(OA\perp TA\) and \(OB\perp TB\), triangles \(OTA\) and \(OTB\) are right triangles. In them, \(OA=OB\) as radii, and \(OT\) is the common hypotenuse. Hence the triangles are congruent. Therefore \(TA=TB\), and also \(\angle ATO=\angle OTB\), so \(OT\) bisects angle \(ATB\).
Although “tangents from one point are equal” is not the main topic, it is a natural consequence of the radius-tangent perpendicularity.