Tangent lines to circles

In Euclidean plane geometry, tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

Tangent lines to one circle

A tangent line "t" to a circle "C" intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed.

The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius.

No tangent line can be drawn through a point in the interior of a circle, since any such line must be a secant line. However, "two" tangent lines can be drawn to a circle from a point P outside of the circle. The geometrical figure of circle and both tangent lines likewise have a reflection symmetry about the radial axis joining P to the center point O of the circle. Thus length of the segments from P to the two tangent points are equal. 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. The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency.

Geometrical constructions

It is relatively straightforward to construct a line "t" tangent to a circle at a point T on the circumference of the circle. A line "a" is drawn from O, the center of the circle, through the radial point T; the line "t" is the perpendicular line to "a". One method for constructing this perpendicular isas follows. Placing the compass point on T with the circle's radius "r", a second point G is identified on the radial line "a"; thus, T is the midpoint of the line segment OG. Two intersecting circles of the same radius "R">"r" are drawn, centered on O and G, respectively. The line drawn through their two points of intersection is the tangent line.

Thales' theorem may be used to construct the tangent lines to a point P external to the circle "C". A circle is drawn centered on M, the midpoint of the line segment OP, where O is again the center of the circle "C". The intersection points T₁ and T₂ are the tangent points for lines passing through P, by the following argument. The line segments OT₁ and OT₂ are radii of the circle "C"; since both are inscribed in a semicircle, they are perpendicular to the line segments PT₁ and PT₂, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T₁ and T₂ are tangent to the circle "C".

Tangent quadrilateral theorem and inscribed circles

A tangent 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. The sum of opposite sides of any such quadrilateral are equal, i.e.,

:$overline\{AB\}\; +\; overline\{CD\}\; =\; overline\{BC\}\; +\; overline\{DA\}.$

This conclusion follows from the equality of the tangent segments from the four points of the quadrilateral. Let the tangent points be denoted as P (on segment AB), Q (on segment BC), R (on segment CD) and S (on segment DA). The the symmetric tangent segments about each point of ABCD are equal, e.g., BP=BQ="b", CQ=CR="c", DR=DS="d", and AS=

:$overline\{AB\}\; +\; overline\{CD\}\; =\; (a+b)\; +\; (c+d)\; =\; overline\{BC\}\; +\; overline\{DA\}\; =\; (b+c)\; +\; (d+a)$ proving the theorem. 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.

This theorem has various uses. For example, it shows immediately that no rectangle can have an inscribed circle unless it is a square, and that every rhombus has an inscribed circle, whereas a general parallelogram does not.

Tangent lines to two circles

[
homothetic centers of the two circles (red) are shown as black points.]

For two circles, there are generally four lines that are tangent to both. For two of these, the external tangent lines, the circles fall on the same side of the line; for the two others, the internal tangent lines, the circles fall on opposite sides of the line. The external tangent lines intersect in the external homothetic center, whereas the internal tangent lines intersect at the internal homothetic center. The internal homothetic center always lies on the line joining the centers of the two circles, in the segment between the two circles. The exterior center lies on the same line, and is closer to the center of the smaller than to the center of the larger. If the two circles have equal radius, the external tangent lines are parallel, and—at least in inversive geometry— the external homothetic center, lies at infinity. [cite web|url=http://whistleralley.com/tangents/tangents.htm|accessdate=2008-09-29|title=Tangent circles|publisher=Whistleralley.com|author=Paul Kunkel]

Geometrical constructions

Let O₁ and O₂ be the centers of the two circles, "C"₁ and "C"₂ and let "r"₁ and "r"₂ be their radii, with"r"₁ > "r"₂; in other words, circle "C"₁ is defined as the larger of the two circles. Two different methods may be used to construct the external and internal tangent lines.

;External tangents

A new circle "C"₃ of radius "r"₁ − "r"₂ is drawn centered on O₁. Using the method above, two lines are drawn from O₂ that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking both circles "C"₁ and "C"₂ by a constant amount, "r"₂, which shrinks "C"₂ to a point. Two radial lines may be drawn from the center O₁ through the tangent points on "C"₃; these intersect "C"₁ at the desired tangent points. The desired external tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above.

;Internal tangents

A new circle "C"₃ of radius "r"₁ + "r"₂ is drawn centered on O₁. Using the method above, two lines are drawn from O₂ that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking "C"₂ to a point while expanding "C"₁ by a constant amount, "r"₂. Two radial lines may be drawn from the center O₁ through the tangent points on "C"₃; these intersect "C"₁ at the desired tangent points. The desired internal tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above.

Algebraic solutions

Belt problem

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. Conversely, if the belt is wrapped exteriorly around the pulleys, the exterior tangent line segments are relevant; this case is sometimes called the "pulley problem".

Illustration

In general the points of tangency "t"₁ and "t"₂ for the four lines tangent to two circles with centers "x"₁ and "x"₂ and radii "r"₁ and "r"₂ are given by solving the simultaneous equations:

:
(t_1 - x_1)(t_1 - x_1)| & = r_1^2 \\
(t_2 - x_2)(t_2 - x_2)| & = r_2^2 end{align}

Outer tangent

The red line joining the points (x3,y3) and (x4,y4) is the outer tangent between the two circles. Given points (x1,y1), (x2,y2) the points (x3,y3), (x4,y4) can easily be calculated by equating the angle theta and adding the x,y coordinates of the triangle(theta) to the original coordinates (x1,y1) as shown in the figure.

Inner tangent

An inner tangent is a tangent that intersects the segment joining two circles' centers. Note that the inner tangent will not be defined for cases when the two circles overlap.

Tangent lines to three circles: Monge's theorem

For three circles denoted by "C"₁, "C"₂, and "C"₃, there are three pairs of circles ("C"₁"C"₂, "C"₂"C"₃, and "C"₁"C"₃). Since each pair of circles has two homothetic centers, there are six homothetic centers altogether. Gaspard Monge showed in the early 19th century that these six points lie on four lines, each line having three collinear points.

Problem of Apollonius

Many special cases of Apollonius' 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). To solve this problem, the center of any such circle must lie on an angle bisector of the any pair of the lines; there are two angle-bisecting lines for every intersection of two lines. The intersections of these angle bisectors give the centers of solution circles. There are four such circles in general, the inscribed circle of the triangle formed by the intersection of the three lines, and the three exscribed 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). To accomplish this, it suffices to scale two of the three given circles until they just touch, i.e., are tangent. An inversion in their tangent point with respect to a circle of appropriate radius transforms the two touching given circles into two parallel lines, and the third given circle into another circle. Thus, the solutions may be found by sliding a circle of constant radius between two parallel lines until it contacts the transformed third circle. Re-inversion produces the corresponding solutions to the original problem.

Generalizations

The concept of a tangent line to one or more circles can be generalized in several ways. First, the conjugate relationship between tangent points and tangent lines can be generalized to pole points and polar lines, in which the pole points may be anywhere, not only on the circumference of the circle. 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. A generic quartic curve has 28 bitangents.

A third generalization considers tangent circles, rather than tangent lines; a tangent line can be considered as a tangent circle of infinite radius. In particular, the external tangent lines to two circles are limiting cases of a family of circles which are internally or externally tangent to both circles, while the internal tangent lines are limiting cases of a family of circles which are internally tangent to one and externally tangent to the other of the two circles. [citation|first=Paul|last=Kunkel|year=2007|title=The tangency problem of Apollonius: three looks|journal=BSHM Bulletin: Journal of the British Society for the History of Mathematics|volume= 22|issue=1|pages= 34–46|doi=10.1080/17498430601148911|url=http://whistleralley.com/tangents/bshmkunkel.pdf]

In Möbius or inversive geometry, lines are viewed as circles through a point "at infinity" and for any line and any circle, there is a Möbius transformation which maps one to the other. In Möbius geometry, tangency between a line and a circle becomes a special case of tangency between two circles. This equivalence is extended further in Lie sphere geometry.

References

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