Archive for tangent

Circle Theorems

Posted in The Angle of Circle with tags , , , , , , , , , , , , on January 10, 2010 by merrusmsk

Circle Theorems

Circles

A circle is a set of points which are all a certain distance from a fixed point known as the centre.
A line joining the centre of a circle to any of the points on the circle is known as a radius.

The circumference of a circle is the length of the circle. The circumference of a circle = 2 × π × the radius.

a sector, an arc and chord

The red line in the second diagram is called a chord. It divides the circle into a major segment and a minor segment.

Theorems

Angles Subtended on the Same Arc

Angles subtended on the same arc

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.

Angle in a Semi-Circle

angle in a semi-circle

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.

This proof is higher tier Proof

We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

Divide the triangle in two

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

Two isosceles triangles

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.

Tangents

A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).

A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.

angle with a tangent

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

Tangents from an external point are equal in length

Angle at the Centre

Angle at the centre

The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.

This proof is higher tier Proof

You might have to be able to prove this fact:

proof diagram 1

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a
Similarly, ∠OXB = b

proof diagram 2
Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a
Similarly, ∠BOX = 180 – 2b
Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX
= 360 – (180 – 2a) – (180 – 2b)
= 2a + 2b = 2(a + b) = 2 ∠AXB

This section is higher tier Alternate Segment Theorem

Alternate segment theorem

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.

Proof

You may have to be able to prove the alternate segment theorem:

proof of alternate segment theorem

We use facts about related angles:

A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90
But OAC + x = 90, so ∠OAC + x = ∠OAC + y
Hence x = y

Cyclic Quadrilaterals

cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.

This proof is higher tier Area of Sector and Arc Length

A sector

If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360

In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360

Fof more detail just type circle theorem in ur search list..

by

Melson Manggis

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Tangent lines to circles

Posted in Tangent of circle with tags , on December 28, 2009 by msrica

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.

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).

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 angle PTO between a tangent line and a radial line segment is a right angle. By the inscribed angle theorem, the angle PTS between the tangent line and another point on the circumference equals half of the central angle TOS subtended by the two points T and S.

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 is as 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 allows the construction of a tangent line PT to a given circle (solid black). A semicircle with diameter OP intersects the given circle at the desired point T. The other tangent point is the second intersection of the dashed and given circles.

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}.

Tangential quadrilateral

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=AP=a. But each side of the quadrilateral is composed of two such tangent segments

 \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

The external (above) and internal (below) 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.

Geometrical constructions

Let O1 and O2 be the centers of the two circles, C1 and C2 and let r1 and r2 be their radii, with r1r2; in other words, circle C1 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 C3 of radius r1 − r2 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 C1 and C2 by a constant amount, r2, which shrinks C2 to a point. Two radial lines may be drawn from the center O1 through the tangent points on C3; these intersect C1 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 C3 of radius r1r2 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 C2 to a point while expanding C1 by a constant amount, r2. Two radial lines may be drawn from the center O1 through the tangent points on C3; these intersect C1 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

Main article: 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

Circle tangents

In general the points of tangency t1 and t2 for the four lines tangent to two circles with centers x1 and x2 and radii r1 and r2 are given by solving the simultaneous equations:

 \begin{align} (t_2 - x_2)(t_2 - t_1) & = 0 \\ (t_1 - x_1)(t_2 - t_1) & = 0 \\ |(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

Two circles’ outer tangents.

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

The internal tangent lines pass through the internal homothetic center.

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

Main article: Monge’s theorem

For three circles denoted by C1, C2, and C3, there are three pairs of circles (C1C2, C2C3, and C1C3). 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

Main article: Special cases of Apollonius’ problem

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.

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.

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 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.

Main articles: Pole and polar and Inversive geometry

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.

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.

by,

Erica Olivia Henry

How To Construct A Tangent To A Circle

Posted in Tangent of circle with tags , , , on December 27, 2009 by gloria terrence firaon
Step-by-step Instructions  
After doing this Your work should look like this
We start with a point P somewhere on a given circle, with center point O.If the center is not given, you can use: “Finding the center of a circle with compass and straightedge or ruler“,
or
“Finding the center of a circle with any right-angled object”.
Geometry construction with compass and straightedge or ruler or ruler
1. Draw a straight line through the center O of the circle and the point P right across the circle. This is a diameter of the circle. Geometry construction with compass and straightedge or ruler or ruler
2. Mark a point Q anywhere. For best accuracy, avoid putting it too close to the diameter line. Geometry construction with compass and straightedge or ruler or ruler
3. Place the compass on the point Q just drawn, and set it’s width to the point P. Geometry construction with compass and straightedge or ruler or ruler
4.  Without changing the compass width, draw an arc across the diameter line, creating point R. Geometry construction with compass and straightedge or ruler or ruler
5.  Again, without changing the compass width, draw another arc on the opposite side of Q. Geometry construction with compass and straightedge or ruler or ruler
6. Using the straightedge, draw a line through R and Q, extending it onwards so it crosses the arc just drawn. Mark this point S. Geometry construction with compass and straightedge or ruler or ruler
7. Using the straightedge, draw a line through P and S, extending it in both directions. Geometry construction with compass and straightedge or ruler or ruler
8. Done. The line just drawn is the tangent to the circle O through point P. Geometry construction with compass and straightedge or ruler or ruler

Posted by,

Gloria Terrence Firaon

tangent of circle

Posted in Tangent of circle with tags , , , on December 27, 2009 by shuyen22

Tangent

  • The line drawn perpendicular to a radius through the end point of the radius is a tangent to the circle.
  • A line drawn perpendicular to a tangent through the point of contact with a circle passes through the center of the circle.
  • Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

Tangent lines

Main article: Tangent lines to circles

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has center (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1a)x+(y1b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is

(x1a)x + (y1b)y = (x1a)x1 + (y1b)y1

or

(x1a)(xa) + (y1b)(yb) = r2.

This application is controlled by the application menu on top of the screen (figure 1.24) and the selection menu on bottom of the screen (figure 1.18). It is used to build and manipulate 2D objects made up of points, segments, arcs and splines. Circles and straight lines are only used as construction aids.

  • The application uses elementary geometry to define points, straight lines, circles, segments and arcs. For example, by using the tangency property, it is possible to define a circle tangent to three straight lines or a straight line tangent to two circles.
  • The application uses splines—a series of points—to define open or closed curves.
  • The application is able to cut one object with another object, except the circles and straight lines, which are only construction aids. It is for example possible to cut an object by a straight line. However, the straight line will not be affected by this operation.
  • The application is able to round off angles.
  • The application is able to duplicate an object n times by using the classical affine transformations: symmetry, rotation, homothety.
  • The application is able to “outline” figures.
  • The application is able to invert arcs and segments.
  • etc.

 Page II.2 of the exhibition René Descartes
Géométrie

Let CE be the curve, and assume that through point C we need to draw a straight line that forms straight angles with it. I imagine everything already done and assume CP as the sought line, a line that I prolong until P where it meets straight line GA, which I imagine being that to which all the points of line CE must refer. Thus position MA or CB = y, CM or BA = x, I will obtain a certain equation that expresses the relation between x and y. […]

With a familiar language and formalism, Descartes continues by explaining that if one poses for the unknown circle PC=s and PA=v, observing that the triangle PMC has a straight angle, one has $s^2=x^2+v^2-2vy+y^2$, from which one can recover x (or equivalently, y) and substitute it in the equation of the given curve. Then,

“after having found [that equation] instead of using it to know the quantities x or y […] that are already given because point C [in which we must determine the normal to the curve] is given, we must use it to find v or s that determine the requested point P [centre of the sought circle]. For this reason, we must consider that if this point P is the way we want it, the circle of which this is the centre and that will pass through C, will touch the curve CE there without intersecting it. On the other hand, if P is a little closer or a little further from A of that which it should be, the circle will intersect the curve not only in point C but necessarily also in some other [E]. […] but the more these two points, C and E, will be near, the less the difference between the roots [of the equation]. Finally, if these two points are one (that is if the circle that goes through C touches the curve there without intersecting it), the roots will be exactly the same […].”

Therefore it will be sufficient to impose that the polynomial has two double roots. If the equation of the curve was of degree $m$, the resulting polynomial will have degree $2m$ and will be of the form $(y-y_0)^2Q(y)$ where $Q(y)$is a generic polynomial of degree $2(m-1)$. Equalising the coefficients of homologous powers we obtain $2m+1$ equations from which we can get the $2m-1$ coefficients of $Q$ as well as the parameters v and s.

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The Géométrie was diffused among mathematicians mostly through the two Latin editions edited by Franz van Schooten. The first was published in 1649 and, together with the translation of the Géométrie , it contains De Beaune’s Notae Breves and Schooten’s own Comentarii. In the second edition, in two volumes, many other pamphlets are added, among which two letters by John Hudde containing a theorem on double roots that brings to a simplification of the previous method. Schooten’s commentary cover all three books of the Géométrie with precise notes, observations, integrations and applications. Regarding the problem of tangents, Schooten illustrates how to apply Descartes’ method in various examples, among which the determination of the normal to the conchoid.

* Page II.3 in the exhibition

Franz van Schooten
In geometriam Renati Des Cartes Commentarii

Let CE be the first conchoid of the ancients, with a pole G and a centre line AB, such that all segments whose prolongations intersect in G and are comprised between the curve CE and the straight line AB (like AE, LC) are equal. We require to draw a straight line (like CP), that intersects the conchoid at a straight angle in a given point C. […] It is then enough to take on the straight line CG the segment CD equal to CB, which is perpendicular to AB, and then from point D draw DF parallel to AG and equal to GL; thus we will have point F, through which we will draw the required line CP.

[follows the construction with Descartes’ method]

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The kinematic construction of tangents

In 1644 Mersenne divulged a method to draw the tangents to a curve, communicated to him by Gilles Personne de Roberval, professor at Paris’s College Royal. In the same year, Torricelli published a similar method in his Opera Geometrica. Both methods presuppose the knowledge of the kinematic decomposition of the curve of which the tangent must be drawn: in the parabola, for example, a point distances itself from the focus with the same speed with which it distances itself from the centre line, in the ellipse it approaches a focus with the same speed with which it distances itself from the other, in the spiral a point rotates around the origin at the same speed with which it distances itself from it, a fact already known and used by Archimedes. Roberval’s manuscript, written down by a pupil of his, was presented in 1668 to the Académie des Sciences and published in a collection of writings only in 1693. The “axiom or principle of invention” at the basis of the method is that “the direction of movement of a point describing a curved line is the tangent to the curved line in any position of that point”, a principle which is “sufficiently intelligible” that “one will easily accept if one considers it with some attention.” Hence descends the “general rule” to follow for the tangents:

For the specific properties of the curved line (which will be given to you), examine the various movements that the point describing the line has where you want to draw the tangent: compose all these movements in one, trace the line of the direction of the composite movement, and you will have the tangent to the curved line.

By applying the rule “word by word”, one can study several curves:

tangents to conical sections, tangents to the other main curves known to the ancients and to other recently described curves, like Mr. Pascal’s snail, Mr. Roberval’s cycloid, Mr. Descartes’s parabola of the second type, etc.

The eleventh example of the pamphlet deals actually with the cycloid that Robertval calls “roulette” o “trochoïde”. The curve is described by a point B that is on a circumference as this rolls on a straight line BC. Another way of generating the curve is by saying that the circumference translates with an uniform motion so that the centre a describes the base segment, and at the same time point B uniformly traces the circumference. If the length of the basis is the same than the circumference, we have a roulette of the first type, but in general we can consider the cases in which the base is longer or shorter than the circumference. After describing the construction by points of the curve, Roberval describes the construction of the tangent in any point E on the basis of the decomposition into two simultaneous motions.

* Page II.4 in the exhibition

Gilles Personne de Roberval
Observations sur la composition des mouvements et sur le moyen de trouver les touchantes des lignes courbes.

be given; we require its tangent in point E. Describe the circle BDF of the roulette […]; from point E draw the straight line EF parallel to AC which intersects in F the circumference of the roulette’s semicircle […]; draw FG tangent to the circle, then take H on the tangent to the circle so that AC is to the circumference of the circle like EF is to FH; from point H draw HE, and this will be the tangent to the roulette. 

Roberval’s construction is then compared to Fermat’s:

“one draws EF as above, draws the straight line FB and through the point E draws AH parallel to FB. EH will be the tangent”.

It is proven that the two construction agree, but, it is said, Fermat’s method is not as general because it only works for the cycloid of the first type.

by,

Seah Shu Yen