§10.1 · Positions of a line w.r.t. a circle
Non-intersecting, Secant, or Tangent?
Drag the line up and down. Watch how many points it shares with the circle — that count decides its name. Notice how the secant collapses into a tangent the instant its two crossing points merge into one.
Instrument
Tip: you can also click-drag the line directly on the canvas.
A tangent is the special, limiting case of a secant — the case where the two points where it cuts the circle slide together into a single point of contact.
§10.2 · Theorem 10.1
The tangent is perpendicular to the radius
Drag point P around the circle to place the tangent anywhere. Then drag Q along the tangent line away from P. Watch OQ — it is always longer than the radius OP, which is only possible if OP meets the tangent at exactly 90°.
Instrument
The tangent at any point of a circle is perpendicular to the radius through the point of contact. Proof idea: every other point Q on the tangent lies outside the circle, so OQ > OP always. That makes OP the shortest distance from O to the line — and the shortest distance from a point to a line is always perpendicular to it.
§10.3 · Number of tangents from a point
Inside, on, or outside the circle?
Drag point P anywhere on the canvas. The lab automatically draws every possible tangent from P to the circle.
Case detector
No tangent is possible. Every line through an interior point cuts the circle twice.
Exactly one tangent exists — the unique line through that point perpendicular to the radius.
Exactly two tangents can be drawn, touching the circle at points T₁ and T₂.
§10.3 · Theorem 10.2 & angle relations
Equal tangent lengths, and the angle they make
Drag external point T. The two tangent segments TP and TQ are always exactly equal in length — this lab measures them live. It also verifies Example 2: ∠PTQ = 2 × ∠OPQ, for any position of T.
Instrument
The lengths of the two tangents drawn from an external point to a circle are equal (proved using congruent right triangles OTP and OTQ, RHS rule). It also follows that OT bisects ∠PTQ — the centre always lies on the angle bisector of the two tangents.
Applying the theorems
Practice problems
Problem A (a quadrilateral circumscribing a circle): drag each contact point's split to see that AB + CD = AD + BC always holds, because every tangent pair from a vertex is equal.
Tangent segments from each vertex
Problem B (Example 3): PQ is a chord of a circle of radius r. Tangents at P and Q meet at T. Drag the sliders to change r and the chord length, and watch TP recompute live using the similar-triangles method.