Even in Ancient Egypt, science appeared, with the help of which it was possible to measure volumes, areas and other quantities. The impetus for this was the construction of the pyramids. It involved a significant number of complex calculations. And besides construction, it was important to correctly measure the land. This is where the science of "geometry" came from the Greek words "geos" - earth and "metrio" - I measure.
The study of geometric shapes contributed to the observation of astronomical phenomena. And already in the 17th century BC. e. Initial methods were found for calculating the area of the circle, the volume of the ball, and the most important discovery — the Pythagorean theorem.
The statement of the theorem on a circle inscribed in a triangle looks as follows:
Only one circle can be entered in a triangle.
With this arrangement, the circle is inscribed, and the triangle is described around the circle.
The statement of the theorem on the center of a circle inscribed in a triangle is as follows:
The center point of a circle inscribed in a triangle is the intersection point of the bisectors of this triangle.
Circle inscribed in an isosceles triangle
A circle is considered inscribed in a triangle if it touches all its sides with at least one point.
The photo below shows a circle inside an isosceles triangle. The condition of the theorem on a circle inscribed in a triangle is met - it applies to all sides of the triangle AB, BC and CA at points R, S, Q, respectively.
One of the properties of an isosceles triangle is that the inscribed circle touches the base in half (BS = SC), and the radius of the inscribed circle is one third of the height of the triangle (SP = AS / 3).
Properties of the theorem on a circle inscribed in a triangle:
- The segments extending from one vertex of the triangle to the points of tangency with the circle are equal. In the figure, AR = AQ, BR = BS, CS = CQ.
- The radius of the circle (inscribed) is the area divided by the semi-perimeter of the triangle. As an example, you need to draw an isosceles triangle with the same letter designations as in the picture, the following sizes: base BC = 3 cm, height AS = 2 cm, sides AB = BC, respectively, 2.5 cm each. We draw a bisector from each angle and denote the place of their intersection as P. Enter a circle with radius PS, the length of which must be found. You can find out the area of a triangle by multiplying 1/2 of the base by height: S = 1/2 * DC * AS = 1/2 * 3 * 2 = 3 cm 2 . The semi-perimeter of the triangle is 1/2 the sum of all sides: P = (AB + BC + CA) / 2 = (2.5 + 3 + 2.5) / 2 = 4 cm; PS = S / P = 3/4 = 0.75 cm 2 , which is fully true, if measured with a ruler. Accordingly, the property of the theorem on a circle inscribed in a triangle is true.
Circle inscribed in a right triangle.
For a triangle with a right angle, the properties of the inscribed circle theorem in the triangle apply. And, in addition, the ability to solve problems with the postulates of the Pythagorean theorem is added.
The radius of the inscribed circle in a right-angled triangle can be defined as follows: add the legs length, subtract the hypotenuse value and divide the resulting value by 2.
There is a good formula that will help to calculate the area of a triangle - multiply the perimeter by the radius of the circle inscribed in this triangle.
Statement of the inscribed circle theorem
In planimetry, theorems on inscribed and described figures are important. One of them sounds like this:
The center of the circle inscribed in the triangle is the intersection point of the bisectors drawn from its angles.
The presented figure shows the proof of this theorem. The equality of angles is shown, and, accordingly, the equality of adjacent triangles.
Theorem on the center of a circle inscribed in a triangle
The radii of a circle inscribed in a triangle drawn at points of tangency are perpendicular to the sides of the triangle.
The task “formulate a theorem on a circle inscribed in a triangle” should not be taken by surprise, because this is one of the fundamental and simplest knowledge in geometry, which must be fully mastered to solve many practical problems in real life.