GeoProof — Equal Tangents explorer

PT₁ = PT₂ = √(OP² − r²)

Equal Tangents from a Point

From a point P outside a circle there are exactly two tangent lines to the circle. The two tangent segments — from P to each point of contact — are always equal in length:

PT₁ = PT₂ = √(OP² − r²)

Drag P anywhere outside and the two lengths stay matched.

Why it's true

Each radius meets its tangent at a right angle (tangent ⟂ radius). So triangles OT₁P and OT₂P are right triangles that share the hypotenuse OP and have equal legs OT₁ = OT₂ = r. They are congruent, which forces PT₁ = PT₂. The common length is √(OP² − r²) by the Pythagorean theorem.

A line of symmetry

The figure is mirror-symmetric across the line OP: it bisects the angle T₁PT₂ and the angle T₁OT₂. So OP is the angle bisector of the two tangents.

Where it's used

Equal tangents are the reason a circle can be inscribed in a triangle or polygon (each vertex sends two equal tangents to the incircle) — the key step behind Pitot's theorem for tangential quadrilaterals.