Secants tangents - all this hundreds of times you could hear the lessons of geometry.But the release of the school behind, pass the year, and all this knowledge forgotten.What should I remember?
essence
term "tangent to the circle" sign, perhaps, everything.But it is unlikely that all will soon formulate its definition.Meanwhile, this is called the tangent lines lying in the same plane with the circle that intersects it at one point.They may be a great many, but they all have the same properties, which are discussed below.As you might guess, the point of contact referred to the place where the circle and the straight line intersect.In each case, she is one, and if there are more, then it will be transversal.
history of discovery and study
concept tangent appeared in ancient times.The construction of these lines to the circle first, and then to the ellipses, parabolas and hyperbolas with a ruler and compass held still in the early stages of the development of geometry.Of course, history has not preserved the name of the discoverer, but it is evident that even while people were well-known properties of tangent to the circle.
In modern times, the interest in this phenomenon erupted again - began a new round of study of this concept in conjunction with the opening of new curves.Thus, Galileo introduced the concept of cycloid and Farm and Descartes built a tangent to it.As for the circles, it seems, is not left to the ancient secrets in this field.
Properties
radius drawn to the point of intersection is perpendicular to the line.This is the main but not the only feature that is tangent to the circle.Another important feature already includes two straight.Thus, a common point lying outside the circle can be made two tangents, and their lengths are equal.There is another theorem on this subject, but it is rarely held in the framework of the standard school course, but to solve some problems, it is extremely convenient.It goes as follows.From one point located outside the circle, draw a tangent and secant to it.Image of the segment AB, AC and AD.A - the intersection of lines, B point of contact, C and D - the intersection.In this case, it is fair to the following equation: the length of the tangent to the circle, squared, is equal to the product of AC and AD.
From the foregoing, there is an important corollary.For each point of the circle can construct a tangent, but only one.The proof of this is simple: it is theoretically omitting perpendicular from the radius, we find out that form a triangle can not exist.And that means that the tangent - the only one.
Building
Among other tasks in geometry there is a special category, as a rule, do not enjoy the love of pupils and students.To solve the tasks of this category need only ruler and compass.It is the task of building.Do they build on a tangent.
So, given a circle and a point lying outside its borders.And you must navigate through them tangent.How to do it?First of all, you need to spend the interval between the center of circle O and set point.Then, using a compass should divide it in half.To do this, you must specify the radius of the - a little more than half the distance between the center of the original circle and the point.Then you need to build two intersecting arcs.Moreover, the radius from the compass should not be changed, and the center of each circle will be part of the original point, and O, respectively.Places need to connect the intersections of arcs that divide the interval in half.Ask for compass radius equal to this distance.Next to the city center at the intersection to build another circle.It will be based on both the original point, and O. In this case there will be two intersection with this problem in a circle.That they will be points of contact for the initially specified point.
Interesting
It is tangent to the circumference of the building led to the birth of the differential calculus.The first work on this subject was published by the famous German mathematician Leibniz.It provided for the possibility of finding the maxima, minima and tangents, regardless of the fractional and irrational quantities.Well, now it is used for many other calculations.
Moreover, the tangent to the circle associated with the geometric tangent sense.It is from this, and its name comes.In Latin tangens - "tangent".Thus, this concept is not only a geometry and differential calculus, but with trigonometry.
Two circles
not always tangent zatragivet only one figure.If one of the circle can hold a great many lines, then why can not the other way around?Can.That's just the problem in this case is seriously complicated, because the tangent to the two circles can not pass through any point, and the relative position of all of these figures can be very different.
types and varieties
When it comes to the two circles, and one or more direct, even if you know that it's about, is not immediately clear how all these figures are in relation to each other.Based on this, there are several varieties.Thus, the circles may have one or two points in common, or none at all.In the first case, they will overlap, and the second - to touch.And here are two varieties.If one circle, as it were embedded in the second, it is called an internal touch - if not something external.To understand the relative position of the pieces is possible not only on the basis of the drawing, and having information about the sum of their radii and the distance between their centers.If these two values are equal, the circles touch.If the first more - intersect and otherwise - have no common points.
So it is with straight lines.For any two circles that do not have common points, it is possible to build four
tangents.Two of them will overlap between the figures, they are called internal.A couple of other - external.
If we are talking about circles, which have one point in common, the problem seriously simplified.The fact that in any mutual position in this case they will be tangent only one.And it will pass through the point of intersection.So that the construction will not cause difficulties.
If the figures have two points of intersection, then they can be constructed line tangent to the circle, as one, and the second, but only outside.Solving this problem is similar to what is discussed later.
Problem Solving
Both internal and external tangent to the two circles in the building are not so simple, though, and the problem is solved.The fact that it uses an auxiliary figure so figured out such a method alone is problematic.Thus, given two circles of different radii and centers O1 and O2.For them, the need to build two pairs of tangents.
First of all, near the center of the larger circle to build supportive.Thus on the compass must be set the difference between the radii of the two original figures.From the center of the smaller circle constructed tangent to the auxiliary.After that of O1 and O2 are held perependikulyary these direct to the intersection with the original figures.As follows from the basic properties of tangent, the required points on both circles found.The problem is solved, at least the first part.
order to build internal tangents have to solve almost a similar problem.Again, we need an auxiliary figure, but this time its radius is equal to the sum of the original.To her construct tangent from the center of one of these circles.The further course of the decision can be understood from the previous example.
tangent to the circle, or even two or more - not such a difficult task.Of course, mathematicians have long ceased to solve similar problems manually and trust calculate special programs.But do not think that it is now not necessarily be able to do it yourself, because for a correct formulation of the task for a computer to do much and understand.Unfortunately, there are fears that after the final transition to the test form of control of knowledge problems on construction will cause the students all the more difficult.
As for finding common tangent to more circles, it is not always possible, even if they lie in the same plane.But in some cases it is possible to find such a line.
life examples
common tangent to the two circles is often found in practice, though it is not always visible.Conveyors, block system, transmission belts pulleys, thread tension in sewing machine, but even just a bicycle chain - are all examples of life.So do not think that geometrical problems remain only in theory: in engineering, physics, construction and many other areas they find practical application.
