Tangent lines to circles

Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs.

This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections.

The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius.

By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle C. This power equals the product of distances from P to any two intersection points of the circle with a secant line passing through P. The tangent line t and the tangent point T have a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines.

A tangential quadrilateral ABCD is a closed figure of four straight sides that are tangent to a given circle C. Equivalently, the circle C is inscribed in the quadrilateral ABCD.

By the Pitot theorem, the sums of opposite sides of any such quadrilateral are equal, i.e.,

The converse is also true: a circle can be inscribed into every quadrilateral in which the lengths of opposite sides sum to the same value.

The external tangent lines intersect in the external homothetic center, whereas the internal tangent lines intersect at the internal homothetic center.

If the two circles have equal radius, there are still four bitangents, but the external tangent lines are parallel and there is no external center in the affine plane; in the projective plane, the external homothetic center lies at the point at infinity corresponding to the slope of these lines.

[3] The red line joining the points (x3, y3) and (x4, y4) is the outer tangent between the two circles.

Here R and r notate the radii of the two circles and the angle α can be computed using basic trigonometry.

The distances between the centers of the nearer and farther circles, O2 and O1 and the point where the two outer tangents of the two circles intersect (homothetic center), S respectively can be found out using similarity as follows:

Here, r can be r1 or r2 depending upon the need to find distances from the centers of the nearer or farther circle, O2 and O1.

Note that the inner tangent will not be defined for cases when the two circles overlap.

Note that in degenerate cases these constructions break down; to simplify exposition this is not discussed in this section, but a form of the construction can work in limit cases (e.g., two circles tangent at one point).

The angle is computed by computing the trigonometric functions of a right triangle whose vertices are the (external) homothetic center, a center of a circle, and a tangent point; the hypotenuse lies on the tangent line, the radius is opposite the angle, and the adjacent side lies on the line of centers.

In general the points of tangency t1 and t2 for the four lines tangent to two circles with centers v1 and v2 and radii r1 and r2 are given by solving the simultaneous equations:

is perpendicular to the radii, and that the tangent points lie on their respective circles.

These are four quadratic equations in two two-dimensional vector variables, and in general position will have four pairs of solutions.

Two distinct circles may have between zero and four bitangent lines, depending on configuration; these can be classified in terms of the distance between the centers and the radii.

Further, the notion of bitangent lines can be extended to circles with negative radius (the same locus of points,

The internal and external tangent lines are useful in solving the belt problem, which is to calculate the length of a belt or rope needed to fit snugly over two pulleys.

If the belt is considered to be a mathematical line of negligible thickness, and if both pulleys are assumed to lie in exactly the same plane, the problem devolves to summing the lengths of the relevant tangent line segments with the lengths of circular arcs subtended by the belt.

If the belt is wrapped about the wheels so as to cross, the interior tangent line segments are relevant.

Many special cases of Apollonius's problem involve finding a circle that is tangent to one or more lines.

The simplest of these is to construct circles that are tangent to three given lines (the LLL problem).

The intersections of these angle bisectors give the centers of solution circles.

A general Apollonius problem can be transformed into the simpler problem of circle tangent to one circle and two parallel lines (itself a special case of the LLC special case).

Second, the union of two circles is a special (reducible) case of a quartic plane curve, and the external and internal tangent lines are the bitangents to this quartic curve.

The parametric representation of the unit hyperbola via radius vector is p(a) = (cosh a, sinh a).

By the power-of-a-point theorem , the product of lengths PM · PN for any ray PMN equals to the square of PT , the length of the tangent line segment (red).
The angle θ between a chord and a tangent is half the arc belonging to the chord.
Construction of a tangent to a given circle (black) from a given exterior point ( P ).
Tangents through a point
Tangential quadrilateral
The external (above) and internal (below) homothetic center S of the two circles.
Finding outer tangent. Two circles' outer tangents.
Outer tangents to two circles
Inner tangent. The external tangent lines pass through the internal homothetic center.
Construction of the outer tangent
Construction of the inner tangent
Finding outer tangent. Circle tangents.
Animation showing the inversive transformation of an Apollonius problem. The blue and red circles swell to tangency, and are inverted in the grey circle, producing two straight lines. The yellow solutions are found by sliding a circle between them until it touches the transformed green circle from within or without.
The concept of a tangent line and tangent point can be generalized to a pole point Q and its corresponding polar line q . The points P and Q are inverses of each other with respect to the circle.