Tangents to a circle · Plane geometry I

A tangent line touches a circle at one point and, from the outside, it always comes in pairs. Those two facts - one right angle, one pair of equal segments - are enough to settle a surprising number of length problems without a single computation.

The tangent and the radius

Theorem

A line is tangent to a circle \(k(O, r)\) at \(T\) if and only if it meets \(k\) at \(T\) and \(OT \perp\) the line. Equivalently, the distance from \(O\) to the line equals \(r\).

Theorem

From a point \(P\) outside \(k\) there are exactly two tangents, touching at \(T_1\) and \(T_2\), and

PT_1 = PT_2 , \qquad \angle OPT_1 = \angle OPT_2 .

Indeed the right triangles \(OT_1P\) and \(OT_2P\) share the hypotenuse \(OP\) and have equal legs \(OT_1 = OT_2 = r\), so they are congruent (RHS).

Two tangents from one external point: equal segments, equal angles at \(P\), and two right angles at the touch points.

Tangent lengths in a triangle

Example

The incircle of triangle \(ABC\) touches \(BC\), \(CA\), \(AB\) at \(D\), \(E\), \(F\). Write \(x = AE = AF\), \(y = BF = BD\), \(z = CD = CE\) - equal tangent segments from each vertex. Then

y + z = a, \qquad z + x = b, \qquad x + y = c .

Adding all three gives \(x + y + z = s\), the semiperimeter, and subtracting one equation at a time,

x = s - a, \qquad y = s - b, \qquad z = s - c .

Principle

Whenever a circle touches several lines of a figure, name the tangent lengths from each vertex by a single letter. Every side of the figure then becomes a sum of two letters, and a question about lengths turns into linear algebra with three unknowns.

The angle between a tangent and a chord

Theorem

Let \(TA\) be a chord of a circle and \(t\) the tangent at \(T\). The angle between \(t\) and \(TA\) equals the inscribed angle \(\angle TBA\) for any point \(B\) of the arc on the far side of \(TA\).

The quickest way to see it: the angle between the tangent and the chord and the inscribed angle from the far arc both equal half the central angle \(\angle TOA\). The tangent behaves like the limiting position of a chord \(TB\) as \(B\) slides into \(T\) - the theorem is the inscribed angle theorem at that limit.

Warm-up

Example

A circle is inscribed in a right triangle with legs \(a\), \(b\) and hypotenuse \(c\). Prove that its radius is \(r = \tfrac{1}{2}(a + b - c)\).

What is special about the vertex with the right angle?

Two tangent segments leave it, and the two radii to the touch points are perpendicular to the legs. Look at the quadrilateral they bound.

It is a square

That quadrilateral has three right angles and two adjacent sides equal to \(r\), so it is a square of side \(r\): the tangent length from the right-angle vertex equals \(r\).

Now use the formula for the tangent length

The tangent length from a vertex is \(s\) minus the opposite side, so with the right angle at \(C\) it equals \(s - c = \tfrac{a+b+c}{2} - c = \tfrac{a+b-c}{2}\). That is \(r\).