Draw Line Through Outside Point of 2 Circles

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 of import function in many geometrical constructions and proofs. Since the tangent line to a circumvolve 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 [edit]

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

The radius of a circumvolve is perpendicular to the tangent line through its endpoint on the circumvolve's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circumvolve and tangent line has a reflection symmetry well-nigh the centrality of the radius.

By the power-of-a-point theorem, the product of lengths PM·PN for any ray PMN equals to the foursquare of PT, the length of the tangent line segment (red).

No tangent line tin be drawn through a signal within a circle, since any such line must exist a secant line. However, two tangent lines tin can exist drawn to a circle from a point P outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the centre signal O of the circle. Thus the lengths 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 angle θ between a chord and a tangent is half the arc belonging to the chord.

The tangent line t and the tangent point T accept a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines. The aforementioned reciprocal relation exists betwixt a point P outside the circle and the secant line joining its two points of tangency.

If a signal P is outside to a circle with center O, and if the tangent lines from P affect the circle at points T and S, so ∠TPS and ∠TOS are supplementary (sum to 180°).

If a chord TM is drawn from the tangency point T of exterior point P and ∠PTM ≤ 90° then ∠PTM = (1/2)∠TOM.

Equation of the tangent line with coordinate [edit]

Suppose that the equation of the circumvolve is ${\displaystyle (ten-a)^{2}+(y-b)^{2}=r^{2}}$ with center at ${\displaystyle (a,b)}$ . Then the tangent line of the circle at ${\displaystyle (x_{ane},y_{i})}$ is

${\displaystyle (x-x_{1})(x_{1}-a)+(y-y_{1})(y_{i}-b)=0}$

This tin exist proved by taking the implicit derivative of the circumvolve.

Compass and straightedge constructions [edit]

It is relatively straightforward to construct a line t tangent to a circumvolve at a bespeak T on the circumference of the circle:

• A line a is drawn from O, the center of the circle, through the radial signal T;
• The line t is the perpendicular line to a.

Construction of a tangent to a given circle (black) from a given exterior point (P).

Thales' theorem may exist used to construct the tangent lines to a betoken P external to the circumvolve C:

• A circumvolve is drawn centered on the midpoint of the line segment OP, having diameter OP, where O is over again the center of the circle C.
• The intersection points T _one and T ₂ of the circle C and the new circle are the tangent points for lines passing through P, by the following argument.

The line segments OT_ane and OT_ii are radii of the circle C; since both are inscribed in a semicircle, they are perpendicular to the line segments PT₁ and PT_two, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T _i and T ₂ are tangent to the circle C.

Another method to construct the tangent lines to a indicate P external to the circle using merely a straightedge:

• Describe whatsoever three different lines through the given point P that intersect the circumvolve twice.
• Let ${\displaystyle A_{1},A_{2},B_{1},B_{2},C_{i},C_{2}}$ be the 6 intersection points, with the same letter corresponding to the same line and the index 1 corresponding to the bespeak closer to P.
• Let D be the signal where the lines ${\displaystyle A_{1}B_{2}}$ and ${\displaystyle A_{two}B_{ane}}$ intersect,
• Similarly E for the lines ${\displaystyle B_{1}C_{2}}$ and ${\displaystyle B_{2}C_{1}}$ .
• Depict a line through D and Due east.
• This line meets the circle at 2 points, F and G.
• The tangents are the lines PF and PG.^[1]

With analytic geometry [edit]

Let ${\displaystyle P=(a,b)}$ exist a point of the circle with equation ${\displaystyle x^{2}+y^{2}=r^{two}}$ . The tangent at ${\displaystyle P}$ has equation ${\displaystyle ax+by=r^{ii}}$ , because ${\displaystyle P}$ lies on both the curves and ${\displaystyle {\vec {OP}}=(a,b)^{T}}$ is a normal vector of the line. The tangent intersects the x-centrality at point ${\displaystyle P_{0}=(x_{0},0)}$ with ${\displaystyle ax_{0}=r^{two}}$ .

Conversely, if one starts with indicate ${\displaystyle P_{0}=(x_{0},0)}$ , than the two tangents through ${\displaystyle P_{0}}$ meet the circle at the two points ${\displaystyle P_{1/2}=(a,b_{\pm })}$ with

${\displaystyle a={\frac {r^{ii}}{x_{0}}},\qquad b_{\pm }=\pm {\sqrt {r^{2}-a^{2}}}=\pm {\frac {r}{x_{0}}}{\sqrt {x_{0}^{ii}-r^{2}}}\quad }$ . Written in vector form:

${\displaystyle {\binom {a}{b_{\pm }}}={\frac {r^{two}}{x_{0}}}{\binom {1}{0}}\pm {\frac {r}{x_{0}}}{\sqrt {x_{0}^{2}-r^{2}}}{\binom {0}{1}}\ .}$

If point ${\displaystyle P_{0}=(x_{0},y_{0})}$ lies not on the x-axis: In the vector form one replaces ${\displaystyle x_{0}}$ by the distance ${\displaystyle \ d_{0}={\sqrt {x_{0}^{ii}+y_{0}^{ii}}}\ }$ and the unit base vectors by the orthogonal unit vectors ${\displaystyle \ {\vec {e}}_{1}={\frac {1}{d_{0}}}{\binom {x_{0}}{y_{0}}},\ {\vec {e}}_{ii}={\frac {1}{d_{0}}}{\binom {-y_{0}}{x_{0}}}\ }$ . Then the tangents through point ${\displaystyle P_{0}}$ bear upon the circle at the points

${\displaystyle {\binom {x_{1/2}}{y_{i/2}}}={\frac {r^{ii}}{d_{0}^{2}}}{\binom {x_{0}}{y_{0}}}\pm {\frac {r}{d_{0}^{2}}}{\sqrt {d_{0}^{2}-r^{2}}}{\binom {-y_{0}}{x_{0}}}\ .}$

For ${\displaystyle d_{0}<r}$ no tangents be.
For ${\displaystyle d_{0}=r}$ point ${\displaystyle P_{0}}$ lies on the circle and in that location is just one tangent with equation ${\displaystyle x_{0}x+y_{0}y=r^{two}}$ .
In case of ${\displaystyle d_{0}>r}$ there are two tangents with equations ${\displaystyle \ x_{ane}ten+y_{ane}y=r^{2},\ x_{two}x+y_{2}y=r^{2}}$ .

Relation to circle inversion: Equation ${\displaystyle ax_{0}=r^{2}}$ describes the circle inversion of point ${\displaystyle (x_{0},0)}$ .

Relation to pole and polar: The polar of point ${\displaystyle (x_{0},0)}$ has equation ${\displaystyle xx_{0}=r^{2}}$ .

Tangential polygons [edit]

A tangential polygon is a polygon each of whose sides is tangent to a item circle, called its incircle. Every triangle is a tangential polygon, as is every regular polygon of whatever number of sides; in improver, for every number of polygon sides there are an infinite number of non-congruent tangential polygons.

Tangent quadrilateral theorem and inscribed circles [edit]

A tangential quadrilateral ABCD is a closed figure of four direct sides that are tangent to a given circle C. Equivalently, the circle C is inscribed in the quadrilateral ABCD. Past the Pitot theorem, the sums of opposite sides of whatsoever such quadrilateral are equal, i.e.,

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

This decision follows from the equality of the tangent segments from the four vertices of the quadrilateral. Permit the tangent points be denoted equally P (on segment AB), Q (on segment BC), R (on segment CD) and S (on segment DA). The symmetric tangent segments about each bespeak of ABCD are equal, e.thou., BP=BQ=b, CQ=CR=c, DR=DS=d, and As=AP=a. But each side of the quadrilateral is equanimous of ii such tangent segments

${\displaystyle {\overline {AB}}+{\overline {CD}}=(a+b)+(c+d)={\overline {BC}}+{\overline {DA}}=(b+c)+(d+a)}$

proving the theorem.

The antipodal is also true: a circumvolve can be inscribed into every quadrilateral in which the lengths of contrary sides sum to the same value.^[two]

This theorem and its converse have various uses. For example, they show immediately that no rectangle can have an inscribed circle unless information technology is a square, and that every rhombus has an inscribed circle, whereas a general parallelogram does not.

Tangent lines to two circles [edit]

For 2 circles, there are generally iv distinct lines that are tangent to both (bitangent) – if the ii circles are outside each other – simply 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 contrary sides of the line. The external tangent lines intersect in the external homothetic center, whereas the internal tangent lines intersect at the internal homothetic eye. Both the external and internal homothetic centers prevarication on the line of centers (the line connecting the centers of the ii circles), closer to the center of the smaller circumvolve: the internal centre is in the segment between the ii circles, while the external center is non between the points, but rather exterior, on the side of the center of the smaller circle. If the two circles have equal radius, there are still four bitangents, just the external tangent lines are parallel and in that location is no external centre in the affine plane; in the projective airplane, the external homothetic center lies at the betoken at infinity corresponding to the gradient of these lines.^[iii]

Outer tangent [edit]

Finding outer tangent. Ii circles' outer tangents.

The cerise line joining the points ${\displaystyle (x_{iii},y_{3})}$ and ${\displaystyle (x_{4},y_{4})}$ is the outer tangent betwixt the two circles. Given points ${\displaystyle (x_{ane},y_{1})}$ , ${\displaystyle (x_{2},y_{2})}$ the points ${\displaystyle (x_{3},y_{3})}$ , ${\displaystyle (x_{4},y_{4})}$ can easily exist calculated with help of the angle ${\displaystyle \alpha }$ :

${\displaystyle {\brainstorm{aligned}x_{three}&=x_{1}\pm r\sin \alpha \\y_{3}&=y_{i}\pm r\cos \alpha \\x_{4}&=x_{2}\pm R\sin \alpha \\y_{4}&=y_{ii}\pm R\cos \alpha \\\end{aligned}}}$

Here R and r notate the radii of the two circles and the angle ${\displaystyle \alpha }$ can be computed using bones trigonometry. You have ${\displaystyle \alpha =\gamma -\beta }$ with ${\displaystyle \gamma =-\arctan \left({\tfrac {y_{ii}-y_{i}}{x_{2}-x_{one}}}\correct)}$ and ${\displaystyle \beta =\pm \arcsin \left({\tfrac {R-r}{\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{two}}}}\correct)}$ . ^{[4] [ failed verification – see give-and-take]}

Inner tangent [edit]

Inner tangent. The external tangent lines laissez passer through the internal homothetic center.

An inner tangent is a tangent that intersects the segment joining two circles' centers. Notation that the inner tangent volition not be divers for cases when the two circles overlap.

Construction [edit]

The bitangent lines can be synthetic either past amalgam the homothetic centers, every bit described at that article, and then amalgam 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 also. Alternatively, the tangent lines and tangent points can be constructed more directly, every bit detailed below. Note that in degenerate cases these constructions break downward; to simplify exposition this is not discussed in this section, just a form of the structure can work in limit cases (e.g., ii circles tangent at 1 bespeak).

Synthetic geometry [edit]

Let O _i and O ₂ be the centers of the ii circles, C ₁ and C _two and let r ₁ and r ₂ be their radii, with r ₁ >r _two; in other words, circle C ₁ is defined as the larger of the ii circles. Two unlike methods may be used to construct the external and internal tangent lines.

External tangents

Construction of the outer tangent

A new circle C _iii of radius r ₁ −r _ii is drawn centered on O ₁. Using the method higher up, two lines are drawn from O ₂ that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the state of affairs corresponds to shrinking both circles C ₁ and C ₂ past a constant amount, r ₂, which shrinks C _ii to a point. Two radial lines may be drawn from the center O _one through the tangent points on C ₃; these intersect C _ane 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 synthetic every bit described to a higher place.

Internal tangents

Construction of the inner tangent

A new circle C _iii of radius r ₁ +r ₂ is drawn centered on O ₁. Using the method to a higher place, 2 lines are fatigued from O ₂ that are tangent to this new circumvolve. These lines are parallel to the desired tangent lines, because the state of affairs corresponds to shrinking C ₂ to a bespeak while expanding C ₁ past a constant corporeality, r _ii. Two radial lines may be drawn from the centre O _i through the tangent points on C ₃; these intersect C _i 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 synthetic as described in a higher place.

Analytic geometry [edit]

Let the circles have centres c _one = (x _ane,y ₁) and c _two = (x _two,y _two) with radius r ₁ and r ₂ respectively. Expressing a line by the equation ${\displaystyle ax+by+c=0,}$ with the normalization a ⁱⁱ +b ⁱⁱ = ane, and so a bitangent line satisfies:

ax ₁ +past ₁ +c =r _one and

ax _two +past _two +c =r ₂.

Solving for ${\displaystyle (a,b,c)}$ past subtracting the starting time from the second yields

aΔx +bΔy = Δr

where Δx =x ₂ −x ₁, Δy =y ₂ −y _one and Δr =r ₂ −r ₁.

If ${\displaystyle d={\sqrt {(\Delta ten)^{2}+(\Delta y)^{two}}}}$ is the distance from c ₁ to c _ii we tin can normalize by X = Δx/d, Y = Δy/d and R = Δr/d to simplify equations, yielding the equations aX +past =R and a ⁱⁱ +b ⁱⁱ = 1, solve these to become two solutions (k = ±i) for the ii external tangent lines:

a =RX −kY√(1 −R ⁱⁱ)

b =RY +kX√(1 −R ²)

c =r ₁ − (ax ₁ +past _i)

Geometrically this corresponds to computing the angle formed by the tangent lines and the line of centers, and so using that to rotate the equation for the line of centers to yield an equation for the tangent line. The angle is computed by computing the trigonometric functions of a correct triangle whose vertices are the (external) homothetic centre, a center of a circumvolve, and a tangent point; the hypotenuse lies on the tangent line, the radius is contrary the angle, and the side by side side lies on the line of centers.

(X,Y) is the unit vector pointing from c ₁ to c _ii, while R is ${\displaystyle \cos \theta }$ where ${\displaystyle \theta }$ is the angle between the line of centers and a tangent line. ${\displaystyle \sin \theta }$ is then ${\displaystyle \pm {\sqrt {one-R^{2}}}}$ (depending on the sign of ${\displaystyle \theta }$ , equivalently the management of rotation), and the above equations are rotation of (10,Y) by ${\displaystyle \pm \theta ,}$ using the rotation matrix:

${\displaystyle {\brainstorm{pmatrix}R&\mp {\sqrt {1-R^{2}}}\\\pm {\sqrt {1-R^{2}}}&R\end{pmatrix}}}$

1000 = 1 is the tangent line to the right of the circles looking from c ₁ to c ₂.

yard = −1 is the tangent line to the right of the circles looking from c ₂ to c ₁.

The in a higher place assumes each circle has positive radius. If r ₁ is positive and r _ii negative then c ₁ volition lie to the left of each line and c ₂ to the correct, and the two tangent lines will cross. In this way all four solutions are obtained. Switching signs of both radii switches g = 1 and k = −1.

Vectors [edit]

Finding outer tangent. Circumvolve tangents.

In general the points of tangency t ₁ and t ₂ for the 4 lines tangent to two circles with centers five _ane and v _two and radii r ₁ and r ₂ are given past solving the simultaneous equations:

${\displaystyle {\begin{aligned}(t_{2}-v_{two})\cdot (t_{2}-t_{one})&=0\\(t_{1}-v_{1})\cdot (t_{2}-t_{1})&=0\\(t_{1}-v_{i})\cdot (t_{1}-v_{1})&=r_{1}^{two}\\(t_{2}-v_{2})\cdot (t_{ii}-v_{2})&=r_{2}^{2}\\\stop{aligned}}}$

These equations express that the tangent line, which is parallel to ${\displaystyle t_{two}-t_{1},}$ is perpendicular to the radii, and that the tangent points prevarication on their respective circles.

These are four quadratic equations in 2 two-dimensional vector variables, and in general position will have four pairs of solutions.

Degenerate cases [edit]

Ii singled-out circles may have between zero and iv bitangent lines, depending on configuration; these can be classified in terms of the altitude betwixt the centers and the radii. If counted with multiplicity (counting a mutual tangent twice) at that place are aught, two, or 4 bitangent lines. Bitangent lines tin likewise be generalized to circles with negative or zero radius. The degenerate cases and the multiplicities tin likewise exist understood in terms of limits of other configurations – e.g., a limit of two circles that almost affect, and moving one so that they affect, or a circle with small radius shrinking to a circumvolve of zip radius.

• If the circles are outside each other ( ${\displaystyle d>r_{1}+r_{2}}$ ), which is full general position, there are four bitangents.
• If they touch externally at ane point ( ${\displaystyle d=r_{ane}+r_{2}}$ ) – take one signal of external tangency – then they have two external bitangents and 1 internal bitangent, namely the mutual tangent line. This common tangent line has multiplicity two, every bit it separates the circles (one on the left, 1 on the right) for either orientation (direction).
• If the circles intersect in 2 points ( ${\displaystyle |r_{ane}-r_{two}|<d<r_{1}+r_{ii}}$ ), then they have no internal bitangents and two external bitangents (they cannot be separated, because they intersect, hence no internal bitangents).
• If the circles touch internally at i point ( ${\displaystyle d=|r_{1}-r_{2}|}$ ) – have ane point of internal tangency – and so they have no internal bitangents and ane external bitangent, namely the mutual tangent line, which has multiplicity two, as to a higher place.
• If one circle is completely within the other ( ${\displaystyle d<|r_{one}-r_{2}|}$ ) then they have no bitangents, as a tangent line to the outer circle does non intersect the inner circle, or conversely a tangent line to the inner circle is a secant line to the outer circle.

Finally, if the 2 circles are identical, any tangent to the circle is a common tangent and hence (external) bitangent, and then there is a circumvolve's worth of bitangents.

Farther, the notion of bitangent lines tin exist extended to circles with negative radius (the same locus of points, ${\displaystyle x^{2}+y^{2}=(-r)^{ii},}$ but considered "inside out"), in which case if the radii have opposite sign (one circle has negative radius and the other has positive radius) the external and internal homothetic centers and external and internal bitangents are switched, while if the radii have the same sign (both positive radii or both negative radii) "external" and "internal" accept the aforementioned usual sense (switching one sign switches them, and so switching both switches them back).

Bitangent lines tin also be divers when one or both of the circles has radius zero. In this case the circle with radius zero is a double betoken, and thus any line passing through it intersects the indicate with multiplicity two, hence is "tangent". If one circumvolve has radius cypher, a bitangent line is just a line tangent to the circle and passing through the point, and is counted with multiplicity two. If both circles have radius zero, then the bitangent line is the line they ascertain, and is counted with multiplicity four.

Note that in these degenerate cases the external and internal homothetic center exercise generally nevertheless be (the external middle is at infinity if the radii are equal), except if the circles coincide, in which case the external center is not defined, or if both circles take radius zero, in which case the internal center is not divers.

Applications [edit]

Belt trouble [edit]

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 ii 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 aforementioned plane, the problem devolves to summing the lengths of the relevant tangent line segments with the lengths of circular arcs subtended past the belt. If the belt is wrapped virtually the wheels and then every bit 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.

Tangent lines to 3 circles: Monge's theorem [edit]

For three circles denoted past C ₁, C ₂, and C _iii, at that place are 3 pairs of circles (C _ane C ₂, C _two C ₃, and C _one C ₃). Since each pair of circles has 2 homothetic centers, in that location are six homothetic centers altogether. Gaspard Monge showed in the early on 19th century that these six points prevarication on 4 lines, each line having three collinear points.

Problem of Apollonius [edit]

Blitheness showing the inversive transformation of an Apollonius problem. The bluish and red circles bully to tangency, and are inverted in the grey circumvolve, producing two directly lines. The yellow solutions are found by sliding a circumvolve between them until it touches the transformed green circle from within or without.

Many special cases of Apollonius'due south trouble 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 whatever such circle must lie on an angle bisector of whatever pair of the lines; there are two angle-bisecting lines for every intersection of ii lines. The intersections of these angle bisectors give the centers of solution circles. In that location are iv such circles in general, the inscribed circumvolve of the triangle formed past the intersection of the 3 lines, and the 3 exscribed circles.

A general Apollonius problem can be transformed into the simpler problem of circumvolve tangent to one circumvolve and two parallel lines (itself a special example of the LLC special case). To accomplish this, it suffices to scale two of the three given circles until they just touch, i.due east., are tangent. An inversion in their tangent point with respect to a circle of appropriate radius transforms the ii touching given circles into 2 parallel lines, and the 3rd given circumvolve into some other circle. Thus, the solutions may be found by sliding a circle of constant radius between 2 parallel lines until it contacts the transformed tertiary circle. Re-inversion produces the corresponding solutions to the original problem.

Generalizations [edit]

The concept of a tangent line and tangent bespeak can be generalized to a pole indicate Q and its corresponding polar line q. The points P and Q are inverses of each other with respect to the circumvolve.

The concept of a tangent line to i or more than circles tin exist generalized in several ways. Starting time, the cohabit 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 airplane curve, and the external and internal tangent lines are the bitangents to this quartic bend. 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 circumvolve of space 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 unit of circles which are internally tangent to one and externally tangent to the other of the ii circles.^[5]

In Möbius or inversive geometry, lines are viewed as circles through a indicate "at infinity" and for whatsoever line and whatsoever circle, there is a Möbius transformation which maps one to the other. In Möbius geometry, tangency betwixt a line and a circumvolve becomes a special example of tangency between 2 circles. This equivalence is extended farther in Lie sphere geometry.

Radius and tangent line are perpendicular at a betoken of a circumvolve, and hyperbolic-orthogonal at a point of the unit hyperbola. The parametric representation of the unit hyperbola via radius vector is ${\displaystyle p(a)\ =\ (\cosh a,\sinh a).}$ The derivative of p(a) points in the direction of tangent line at p(a), and is ${\displaystyle {\frac {dp}{da}}\ =\ (\sinh a,\cosh a).}$ The radius and tangent are hyperbolic orthogonal at a since ${\displaystyle p(a)\ {\text{and}}\ {\frac {dp}{da}}}$ are reflections of each other in the asymptote y=x of the unit hyperbola. When interpreted equally split-circuitous numbers (where j j = +ane), the two numbers satisfy ${\displaystyle jp(a)\ =\ {\frac {dp}{da}}.}$

References [edit]

1. ^ "Finding tangents to a circle with a straightedge". Stack Exchange. August fifteen, 2015.
2. ^ Alexander Bogomolny "When A Quadrilateral Is Inscriptible?" at Cut-the-knot
3. ^ Paul Kunkel. "Tangent circles". Whistleralley.com. Retrieved 2008-09-29 .
4. ^ Libeskind, Shlomo (2007), Euclidean and Transformational Geometry: A Deductive Inquiry, pp. 110–112 (online re-create, p. 110, at Google Books)
5. ^ Kunkel, Paul (2007), "The tangency trouble of Apollonius: iii looks" (PDF), BSHM Bulletin: Journal of the British Society for the History of Mathematics, 22 (i): 34–46, doi:10.1080/17498430601148911

External links [edit]

• Weisstein, Eric W. "Tangent lines to 1 circle". MathWorld.
• Weisstein, Eric W. "Tangent lines to ii circles". MathWorld.

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Source: https://en.wikipedia.org/wiki/Tangent_lines_to_circles

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