Advice 1: How to find the incenter

A circle can be inscribed in a corner or in a convex polygon. In the first case, it applies to both sides of the angle in the second — all sides of the polygon. The position of its center in both cases is calculated in similar ways. You will need to conduct geometric constructions.
How to find the incenter
You will need
  • polygon;
  • - the angle of a given size;
  • - a circle with a specified radius;
  • a pair of compasses;
  • - the range;
  • pencil;
  • calculator.
Find the center of the inscribed circle means to determine its position relative to the top individual angle or angles of the polygon. Remember, where is the center of a circleinscribed in a corner. It lies on the bisector. Construct angle of a given size and divide it in half. The radius of the inscribed circle you know. Have inscribed circles he is also the shortest distance from the center to the tangent, i.e. the perpendicular. The tangent in this case is the side angle. Build to one side the perpendicular is equal to the given radius. Ultimate point must lie on the bisector. You got a right triangle. Name it, for example, OSA. O is the vertex of the triangle and the center of the circle, OC is a radius, and OA — cut bisector. The angle CCA is equal to half the original angle. By theorem of sines, find the segment OA, which is the hypotenuse.
To determine the location of the center of the inscribed circle in the polygon and do the same build. Sides of any polygon, by definition, are tangent to the inscribed circle. Accordingly, the radius drawn to any point of tangency will be perpendicular to it. In the triangle center of inscribed circle is the intersection of the bisectors, i.e. the distance from the corners is determined in exactly the same way as in the previous case.
A circle inscribed in a polygon is simultaneously inscribed in every corner. It follows from its definition. Accordingly, the distance of the center from each vertex can be calculated similarly as in the case of a single angle. This is especially important to remember if you're dealing with the wrong polygon. In calculations of the rhombus or square enough to hold the diagonal. The center will coincide with the point of their intersection. To determine its distance from the vertices of the square by using the Pythagorean theorem. In the case of the rhombus is valid the theorem of sines or of the cosines, depending on which angle you are using for calculations.

Advice 2: How to find the radius of the inscribed circle in an equilateral triangle?

Knowing sides of the triangle, find the radius of the inscribed circle. The formula to find the radius and then the circumference and area of a circle, as well as other parameters.
How to find the radius of the inscribed circle in an equilateral triangle?
Imagine an isosceles triangle, which is inscribed in a circle of unknown radius R. Since the circle is inscribed in a triangle, not described around him, all sides of the triangle are tangent to it. The altitude drawn from the vertex of one angle is perpendicular to the base coincides with the median of the triangle. It passes through the radius of the inscribed circle.
It should be noted that isosceles is the triangle whose two sides are equal. The angles at the base of this triangle must also be equal. This triangle at the same time can be inscribed in a circle and describe about her.
First find the unknown base of the triangle. For this purpose, as mentioned above, draw the altitude from the vertex of the triangle to its base. Height crosses the center of the circle. If you know at least one of the sides of a triangle, for instance, side CB, the other party equal because the triangle is isosceles. In this case, it is the side AC. A third party, which is the base of the triangle, find the Pythagorean theorem:
The angle y between the two equal sides way based on the fact that in an isosceles triangle two angles are equal. Accordingly, the third angle y=180-(a+b).
Finding all three sides of the triangle, go to the solution of the problem. The formula connecting the lengths of sides and radius, is as follows:
r=(p-a)(p-b)(p-c)/p, where p=a+b+c/2 - the sum of all sides divided or pauperised.
If circle is inscribed in an isosceles triangle, in this case, it is much easier to find the radius of the circle. With a knowledge of the radius of the circle, you can find important parameters such as the area of a circle and circumference. If the job, on the contrary, given the radius of the circle, and this is in turn a prerequisite to finding the sides of a triangle. Finding sides of a triangle, you can calculate its area and perimeter. These calculations are widely used in many engineering tasks. Planimetry is a basic science, which examine more complicated geometric calculations.
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