In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. 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 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 to a circle intersects the circle at a single point . 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, that 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 within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining to the center point of the circle. Thus the lengths of the segments from to the two tangent points are equal (this is sometimes called the Two Tangents Theorem, see Incircle). By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle . This power equals the product of distances from to any two intersection points of the circle with a secant line passing through .
The tangent line and the tangent point 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 outside the circle and the secant line joining its two points of tangency.
If a point is exterior to a circle with center , and if the tangent lines from touch the circle at points and , then and are supplementary (sum to 180°).
If a chord is drawn from the tangency point of exterior point and then .
Cartesian equation Suppose that the equation of the circle in Cartesian coordinates is with center at . Then the tangent line of the circle at has Cartesian equation
This can be proved by taking the implicit derivative of the circle. Say that the circle has equation of and we are finding the slope of tangent line at where We begin by taking the implicit derivative with respect to :
Now that we have the slope of the tangent line, we can substitute the slope and the coordinate of the tangency point into the line equation .
Compass and straightedge constructions
It is relatively straightforward to construct a line tangent to a circle at a point on the circumference of the circle: A line is drawn from , the center of the circle, through the radial point ; The line is the perpendicular line to .
Thales' theorem may be used to construct the tangent lines to a point external to the circle : A circle is drawn centered on the midpoint of the line segment , having diameter , where is again the center of the circle (cyan). The intersection points and of the circle and the new circle are the tangent points for lines passing through , by the following argument (tan).
The line segments and are radii of the circle ; since both are inscribed in a semicircle, they are perpendicular to the line segments and , respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from and passing through and are tangent to the circle .
Another method to construct the tangent lines to a point external to the circle using only a straightedge:
Draw any three different lines through the given point that intersect the circle twice. Let be the six intersection points, with the same letter corresponding to the same line and the index 1 corresponding to the point closer to . Let be the point where the lines and intersect, Similarly for the lines and . Draw a line through and . This line meets the circle at two points, and . The tangents are the lines and .
With analytic geometry
Let be a point of the circle with equation The tangent at has equation because lies on both the curves and is a normal vector of the line. The tangent intersects the -axis at point with
Conversely, if one starts with point then the two tangents through meet the circle at the two points with
Written in vector form:
If point lies not on the -axis: In the vector form one replaces by the distance and the unit base vectors by the orthogonal unit vectors Then the tangents through point touch the circle at the points
For no tangents exist. For point lies on the circle and there is just one tangent with equation In case of there are 2 tangents with equations
Relation to circle inversion: Equation describes the circle inversion of point
Relation to pole and polar: The polar of point has equation
Tangential polygons
A tangential polygon is a polygon each of whose sides is tangent to a particular circle, called its incircle. Every triangle is a tangential polygon, as is every regular polygon of any number of sides; in addition, for every number of polygon sides there are an infinite number of non-congruent tangential polygons.
Tangent quadrilateral theorem and inscribed circles A tangential quadrilateral is a closed figure of four straight sides that are tangent to a given circle . Equivalently, the circle is inscribed in the quadrilateral . By the Pitot theorem, the sums of opposite sides of any such quadrilateral are equal, i.e.,
This conclusion follows from the equality of the tangent segments from the four vertices of the quadrilateral. Let the tangent points be denoted as (on segment ), (on segment ), (on segment ) and (on segment ). The symmetric tangent segments about each point of are equal:
But each side of the quadrilateral is composed of two such tangent segments
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.Alexander Bogomolny "When A Quadrilateral Is Inscriptible?" at Cut-the-knot
This theorem and its converse have various uses. For example, they show 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
For two circles, there are generally four distinct lines that are tangent to both (bitangent) – if the two circles are outside each other – but in degenerate cases there may be any number between zero and four bitangent lines; these are addressed below. 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. Both the external and internal homothetic centers lie on the line of centers (the line connecting the centers of the two circles), closer to the center of the smaller circle: the internal center is in the segment between the two circles, while the external center is not between the points, but rather outside, on the side of the center of the smaller circle. 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.
Outer tangent
The red line joining the points and is the outer tangent between the two circles. Given points , the points , can easily be calculated with help of the angle :
Here and notate the radii of the two circles and the angle can be computed using basic trigonometry. You have with ()
where atan2 is the 2-argument arctangent.
The distances between the centers of the nearer and farther circles, and and the point where the two outer tangents of the two circles intersect (homothetic center), respectively can be found out using similarity as follows:
Here, can be or depending upon the need to find distances from the centers of the nearer or farther circle, and . is the distance between the centers of two circles.
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.
Construction The bitangent lines can be constructed either by constructing the homothetic centers, as described at that article, and then constructing the tangent lines through the homothetic center that is tangent to one circle, by one of the methods described above. The resulting line will then be tangent to the other circle as well. Alternatively, the tangent lines and tangent points can be constructed more directly, as detailed below. 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).
Synthetic geometry Let and be the centers of the two circles, and and let and be their radii, with ; in other words, circle 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 of radius is drawn centered on . Using the method above, two lines are drawn from that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking both circles and by a constant amount, , which shrinks to a point. Two radial lines may be drawn from the center through the tangent points on ; these intersect 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 of radius is drawn centered on . Using the method above, two lines are drawn from that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking to a point while expanding by a constant amount, . Two radial lines may be drawn from the center through the tangent points on ; these intersect 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.
Analytic geometry Let the circles have centres and with radius and respectively. Expressing a line by the equation with the normalization then a bitangent line satisfies:
Solving for by subtracting the first from the second yields
where
and
for the outer tangent or
for the inner tangent.
If is the distance from to we can normalize by
to simplify equation (1), resulting in the following system of equations:
solve these to get two solutions (; for left tangent, for right tangent) for the two tangent lines ( for outer tangent, for inner tangent):