Among the many subjects of secondary schools there is such as "geometry". It is traditionally believed that the ancestors of this systematic science are the Greeks. Today, Greek geometry is called elementary, since it was she who began the study of the simplest forms: planes, straight lines, regular polygons and triangles. We will stop our attention on the latter, and more precisely on the bisector of this figure. For those who have already forgotten, the bisector of a triangle is a segment of the bisector of one of the corners of the triangle, which divides it in half and connects the vertex with a point located on the opposite side.
The bisector of a triangle has a number of properties that you need to know when solving certain problems:
- The bisector of the angle is the geometrical location of the points that are equally spaced from the sides adjacent to the corner.
- The bisector in the triangle divides the side opposite from the corner into segments that are proportional to the adjacent sides. For example, given a triangle MKB, where a bisector comes out of the angle K connecting the vertex of this angle with point A on the opposite side of MB. Having analyzed this property and our triangle, we have MA / AB = MK / KB.
- The point at which the bisectors of all three angles of the triangle intersect is the center of the circle that is inscribed in the same triangle.
- The base of the bisectors of one external and two internal angles are on the same line, provided that the bisector of the external angle is not parallel to the opposite side of the triangle.
- If two bisectors of one triangle are equal, then this triangle is isosceles.
It should be noted that if three bisectors are given, then constructing a triangle from them, even with the help of a compass, is impossible.
Very often, when solving problems, the bisector of the triangle is unknown, but it is necessary to determine its length. To solve this problem, you need to know the angle that is divided in half by the bisector and the sides adjacent to this angle. In this case, the desired length is defined as the ratio of the double product of the sides adjacent to the corner and the cosine of the angle divided in half to the sum of the sides adjacent to the corner. For example, given the same MKB triangle. The bisector leaves the angle K and intersects the opposite side of the MV at point A. The angle from which the bisector leaves is denoted by y. Now we write down everything that is said in words in the form of a formula: KA = (2 * MK * KB * cos y / 2) / (MK + KB).
If the angle from which the bisector of the triangle comes out is unknown, but all its sides are known, then to calculate the length of the bisector, we will use an additional variable, which we will call the semiperimeter and denote it with the letter P: P = 1/2 * (MK + KB + MB). After that, we will make some changes to the previous formula, which determined the length of the bisector, namely, in the numerator of the fraction we put the double square root of the product of the lengths of the sides adjacent to the corner by the half-meter and quotient, where the length of the third side is subtracted from the half-meter. The denominator is left unchanged. In the form of a formula, it will look like this: KA = 2 * √ (MK * KB * P * (P-MB)) / (MK + KB).
The bisector in a right-angled triangle has all the same properties as in an ordinary, But, besides the already known, there is a new one: the bisectors of the acute angles of a right-angled triangle at an intersection form an angle of 45 degrees. If necessary, this can be easily proved using the properties of a triangle and adjacent angles.
The bisector of an isosceles triangle, along with general properties, has several of its own. Recall what this triangle is. In such a triangle, the two sides are equal, and the angles adjacent to the base are equal. It follows that the bisectors that fall to the sides of an isosceles triangle are equal to each other. In addition, the bisector, lowered to the base, is both the height and the median.