What is the tangent to the circle? Properties of the tangent to the circle. The common tangent to two circles

Secs, tangents - all this hundreds of times you could hear at the lessons of geometry. But the graduation from school behind, pass the years, and all this knowledge is forgotten. What should I remember?

Essence

The term "tangent to the circle" is familiar to everyone, probably. But hardly everyone will be able to quickly formulate his definition. Meanwhile, a tangent is a straight line lying in one plane with a circle that intersects it only at one point. There can be a huge number of them, but they all have the same properties, which we will discuss below. It is not difficult to guess that the point of tangency is the place where the circle and the line intersect. In each case, it is one, but if there are more, it will be already secant.

History of discovery and study

The concept of tangent appeared in ancient times. The construction of these straight lines first to the circle, and then to ellipses, parabolas and hyperbolas with the help of a ruler and a compass was carried out even at the initial stages of the development of geometry. Of course, history did not retain the name of the discoverer, but it is obvious that even at that time people knew the properties of the tangent to the circle.

In modern times, interest in this phenomenon flared up again - a new round of studying this concept began in conjunction with the discovery of new curves. So, Galileo introduced the concept of cycloid, and Fermat and Descartes built a tangent to it. As for the circles, it seems that even for the ancients there were no secrets in this area.

Properties

The radius drawn at the intersection point will be perpendicular to the straight line. it Basic, but not the only property that has a tangent to the circle. Another important feature includes two straight lines. So, through one point lying outside the circle, you can draw two tangents, and their segments will be equal. There is one more theorem on this topic, however, it is rarely held in the framework of a standard school course, although it is extremely convenient for solving some problems. It sounds like this. From a single point located outside the circle, a tangent and a secant are drawn to it. The segments AB, AC and AD are formed. A is the intersection of the lines, B is the point of tangency, C and D are the intersections. In this case, the following equality will be valid: the length of the tangent to the circle, squared, will be equal to the product of the segments AC and AD.

From the above, there is an important consequence. For each point of the circle, one can construct a tangent, but only one. The proof of this is quite simple: theoretically dropping the perpendicular from the radius, we find out that the formed triangle can not exist. And this means that the tangent is unique.

Building

Among other problems in geometry there is a special category, as a rule, not Enjoying the love of students and students. To solve tasks from this category, only the compass and ruler are needed. These are construction tasks. There they are and the construction of a tangent.

So, given a circle and a point lying outside its boundaries. And it is necessary to draw a tangent through them. How can this be done? First of all, we need to draw a segment between the center of the circle O and the given point. Then, using the compass, you should divide it in half. To do this, you need to specify a radius - just over half the distance between the center of the original circle and the given point. After this, we need to construct two intersecting arcs. And the radius of the compass does not need to be changed, and the center of each part of the circle is the initial point and O, respectively. The intersections of the arcs must be joined, which will divide the segment in half. Set a radius equal to this distance on the compass. Further, with the center at the intersection point, construct another circle. It will contain both the original point and O. There will be two more intersections with the given circle in the problem. They will be the points of tangency for the initially specified point.

Interesting

It was the construction of tangents to the circle that led to the birth Differential calculus. The first work on this topic was published by the famous German mathematician Leibniz. He envisaged the possibility of finding maxima, minima, and tangents, regardless of fractional and irrational values. Well, now it is used for many other calculations.

In addition, the tangent to the circle is related to the geometric meaning of the tangent. It is from this that its name derives. In translation from Latin tangens - "tangent". Thus, this concept is associated not only with geometry and differential calculus, but also with trigonometry.

Two circles

Not always the tangent will affect only one figure. If you can draw a huge number of straight lines to one circle, then why not the other way round? Can. That's only the problem in this case is seriously complicated, because the tangent to two circles can not pass through any points, and the mutual arrangement of all these figures can be very Different.

Types and varieties

When it comes to two circles and one or several lines, even if it is known that these are tangent, it does not immediately become clear how all these figures are arranged in relation to each other. On this basis, distinguish several varieties. So, circles can have one or two common points or not have them at all. In the first case, they will intersect, and in the second - touch. And here we distinguish two varieties. If one circle is, as it were, embedded in the second, then the touch is called internal, if not, then external. You can understand the mutual arrangement of figures not only from the drawing, but also with information about the sum of their radii and the distance between their centers. If these two quantities are equal, then the circles touch. If the first is larger - intersect, and if less - then do not have common points.

So it is with straight lines. For any two circles that do not have common points,
Construct four tangents. Two of them will intersect between the figures, they are called internal. A couple of others are external.

If we are talking about circles that have one common point, then the problem is seriously simplified. The fact is that for any mutual arrangement in this case the tangent will only have one. And it will pass through the point of their intersection. So the construction of the difficulty will not cause.

If the figures have two intersection points, a straight line tangent to the circle can be constructed for them, both one and the second, but only the outer one. The solution to this problem is analogous to what will be discussed later.

Problem Solving

Both internal and external tangent to two circles, in construction are not so simple, although this problem is solved. The fact is that an auxiliary figure is used for this, so to think up such a method on your own Is quite problematic. Thus, two circles with different radii and centers O1 and O2 are given. For them, we need to construct two pairs of tangents.

First of all, near the center of a larger circle, we need to construct an auxiliary one. At the same time, the difference between the radii of the two original figures should be established on the compass. From the center of a smaller circle, tangents to the auxiliary circle are constructed. After that, from O1 and O2, perpendiculars are made to these straight lines before crossing with the original figures. As follows from the basic property of the tangent, the required points on both circles are found. The problem is solved, at least, its first part.

In order to construct internal tangents, it is necessary to solve practically A similar problem. Again we need an auxiliary figure, but this time its radius will be equal to the sum of the original ones. To it, tangents are constructed from the center of one of these circles. The further course of the solution can be understood from the previous example.

Tangent to a circle or even two or more is not such a difficult task. Of course, mathematicians have long ceased to solve such problems manually and trust calculations to special programs. But do not think that now you do not need to be able to do it yourself, because to properly formulate tasks for the computer you need to do a lot and understand. Unfortunately, there are fears that after the final transition to the test form of knowledge control, construction tasks will cause more and more difficulties for students.

As for finding common tangents for more circles, this is not always possible, even if they are in the same plane. But in some cases you can find such a straight line.

Examples from life

A common tangent to two circles is often found in practice, although this is not always noticeable. Conveyors, block systems, transfer belts of pulleys, thread tension in a sewing machine, and even just a bicycle chain are all examples from life. So do not think that geometric problems remain only in theory: in engineering, physics, construction and many other areas they find practical application.