Inscribed quadrangle in a circle. Quadrangle ABCD inscribed in a circle

With the division of mathematics into algebra and geometry, learning material becomes more difficult. New figures and their special cases appear. In order to understand the material well, it is necessary to study the concepts, properties of objects and related theorems.

General concepts

By a quadrilateral is meant a geometric figure. It consists of 4 points. Moreover, 3 of them are not located on one straight line. There are segments connecting these points in series.

All quadrangles studied in the school geometry course are shown in the following diagram. Conclusion: any object from the presented figure has the properties of the previous figure.

A quadrangle can be of the following types:

• Parallelogram. The parallelism of its opposite sides is proved by the corresponding theorems.
• Trapezoid. A quadrangle in which the bases are parallel. The other two sides are not.
• Rectangle. A figure in which all 4 angles = 90º.
• Rhombus. A figure in which all sides are equal.
• Square. Combines the properties of the last two figures. He has all sides equal and all angles are right.

The main definition of this topic is an inscribed quadrangle in a circle. It consists in the following. This is the figure around which the circle is described. It must go through all the peaks. The internal corners of a quadrilateral inscribed in a circle add up to 360º.

Not every quadrangle can be inscribed. This is due to the fact that the middle perpendiculars of 4 sides may not intersect at one point. This will make it impossible to find the center of the circle circumscribed around the quadrangle.

Special cases

There are exceptions to every rule. So, in this topic there are also special cases:

• A parallelogram, as such, cannot be inscribed in a circle. Only his special case. This is a rectangle.
• If all the vertices of the rhombus are on the descriptive line, then it is a square.
• All the vertices of the trapezoid are on the boundary of the circle. In this case, they speak of an isosceles figure.

Properties of an inscribed quadrangle in a circle

Before solving simple and complex tasks on a given topic, you need to make sure of your knowledge. Without studying the training material, not a single example can be solved.

Theorem 1

The sum of the opposite angles, the quadrangle inscribed in a circle, is 180º.

Evidence

Given: the quadrilateral ATSD is inscribed in a circle. Its center is point O. It is necessary to prove that _< A + _< C = 180º and _< B + _< D = 180º.

It is necessary to consider the presented figures.

1. _< A is inscribed in a circle centered at the point O. It is measured through ½ BCD (half-arc).
2. _< C is inscribed in the same circle. It is measured through ½ BAD (half-arc).
3. BAD and BCD form a whole circle, i.e., their magnitude is 360º.
4. _< A + _< C are equal to half the sum of the presented arcs.
5. Therefore, _< A + _< C = 360º / 2 = 180º.

In a similar way, the proof for _< B and _< D. occurs. However, there is a second solution to the problem.

1. It is known that the sum of the internal angles of a quadrangle is 360º.
2. Since _< A + _< C = 180º. Accordingly, _< B + _< D = 360º - 180º = 180º.

Theorem 2

(It is often called the inverse) If in a quadrilateral _< A + _< C = 180º and _< B + _< D = 180º (if they are opposite), then a circle can be described around such a figure.

Evidence

Given the sum of the opposite angles of the quadrangle ABCD, equal to 180º. _< A + _< C = 180º, _< B + _< D = 180º. It is necessary to prove that a circle can be described around ABCD.

From the course of geometry it is known that a circle can be drawn through 3 points of a quadrangle. For example, you can use points A, B, C. Where will the point D be located? There are 3 assumptions:

1. She is inside the circle. Moreover, D does not touch the line.
2. Out of the circle. She extends far beyond the boundaries of the outlined line.
3. It turns on the circle.

It should be assumed that D is located inside the circle. The indicated vertex is replaced by D´. It turns out the quadrangle ABCD´.

As a result, it follows: _< B + _< D´ = 2d.

If we continue AD´ to the intersection with the existing circle centered at point E and connect E and C, we get the inscribed quadrangle ABCE. The first theorem implies the equality: _< B + _< E = 2d.

According to the laws of geometry, the expression has no force, since _< D´ is the outer corner of the triangle CD´E. Accordingly, it should be greater than _< E. From this we can conclude that D must be either on the circle or outside it.

In a similar way, it is possible to prove the incorrectness of the third assumption, when D´´ goes beyond the boundary of the described figure.

From two hypotheses follows the only true one. The vertex D is located on a circle line. In other words, D coincides with E. It follows that all points of the quadrangle are located on the line being described.

The following corollaries follow from these two theorems:

• Any rectangle can be inscribed in a circle. There is another consequence. A circle can be described around any rectangle.

• A trapezoid with equal hips can be inscribed in a circle. In other words, it sounds like this: a circle can be described around a trapezoid with equal edges.

A few examples

Problem 1. A quadrangle ABCD is inscribed in a circle. _< ABC = 105º, _< CAD = 35º. Need to find _< ABD. The answer must be written in degrees.

Decision. Initially, it may seem that finding the answer will be difficult.

1. It is necessary to recall the properties of this topic. Namely: the sum of the opposite angles = 180º.

_< ADC = 180º - _< ABC = 180º - 105º = 75º

In geometry, it is better to adhere to the principle: to find everything that is possible. Then come in handy.

2. Next step: use the theorem on the sum of the angles of a triangle.

_< ACD = 180º - _< CAD - _< ADC = 180º - 35º - 75º = 70º

_< ABD and _< ACD are inscribed. By condition, they rely on one arc. Accordingly, they have equal values:

_< ABD = _< ACD = 70º

Answer: _< ABD = 70º.

Problem 2. Dan BCDE is an inscribed quadrangle in a circle. _< B = 69º, _< C = 84º. The center of the circle is point E. Find - _< E.

Decision.

1. It is necessary to find _< E by Theorem 1.

_< E = 180º - _< C = 180º - 84º = 96º

Answer: _< E = 96º.

Problem 3. Given an inscribed quadrangle in a circle. The data are shown in the figure. It is necessary to find unknown quantities x, y, z.

Decision:

z = 180º - 93º = 87º (by Theorem 1)

x = ½ * (58º + 106º) = 82º

y = 180º - 82º = 98º (by Theorem 1)

Answer: z = 87º, x = 82º, y = 98º.

Problem 4. There is an inscribed quadrangle in a circle. Values ​​are shown in the figure. Find x, y.

Decision:

x = 180º - 80º = 100º

y = 180º - 71º = 109º

Answer: x = 100º, y = 109º.

Tasks for an independent solution

Example 1. Given a circle. Its center is point O. AC and BD are diameters. _< ACB = 38º. Need to find _< AOD. The answer must be given in degrees.

Example 2. Given a quadrangle ABCD and a circle circumscribed around it. _< ABC = 110º, _< ABD = 70º. Find _< CAD. Write the answer in degrees.

Example 3. Given a circle and an inscribed quadrangle ABCD. Its two angles are 82º and 58º. You need to find the largest of the remaining angles and record the answer in degrees.

Example 4. Given a quadrangle ABCD. Angles A, B, C are given in a ratio of 1: 2: 3. It is necessary to find the angle D if the specified quadrilateral can be inscribed in a circle. The answer must be given in degrees.

Example 5. Given a quadrangle ABCD. Its sides form arcs of the circumscribed circle. The degrees AB, BC, CD and AD, respectively, are equal: 78˚, 107˚, 39˚, 136˚. Find _< C of the given quadrangle and write down the answer in degrees.