The bisector of the triangle and its properties

Among numerous itemssecondary school there is such as "geometry." Traditionally, it is believed that the ancestors of this systematic science are the Greeks. To date, Greek geometry is called elementary, since it is she who began to study the simplest forms: planes, straight, regular polygons and triangles. On the latter, we will stop our attention, or rather on the bisector of this figure. For those who have already forgotten, the bisector of the triangle is a segment of the bisector of one of the angles of the triangle that divides it in half and connects the vertex to the 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 angle bisector is a geometric locus of points removed at equal distances from the sides adjacent to the corner.
• The bisector in the triangle divides the oppositefrom the angle of the side to the segments, which are proportional to the adjacent sides. For example, a triangle MKB is given, where from the angle K comes a bisectrix connecting the vertex of this angle with the point A on the opposite side MB. Analyzing this property and our triangle, we have MA / AB = MK / KB.
• The point at which the bisectors of all three angles of a triangle intersect is the center of a circle that is inscribed in the same triangle.
• The base of the bisectors of one outer and two inner corners are on the same straight line, provided that the bisector of the outer corner is not parallel to the opposite side of the triangle.
• If two bisectors of the same triangle are equal, then this triangle is isosceles.

It should be noted that if three bisectors are given, then the construction of a triangle over them, even with the help of a compass, is impossible.

Очень часто при решении задач биссектриса triangle is unknown, but it is necessary to determine its length. To solve such a problem it is necessary to know the angle that the bisector divides in half, and the sides adjoining this angle. In this case, the desired length is defined as the ratio of the doubled product of the sides and the cosine of the angle divided by half to the sum of the sides adjacent to the angle. For example, the same MKB triangle is given. The bisector extends from the angle K and crosses the opposite side of the MB at the point A. We denote the angle from which the bisectrix leaves, y. Now let's write down everything that is said in words in the form of a formula: KA = (2 * MK * KB * cos y / 2) / (MK + KB).

Если величина угла, из которого выходит bisector of a triangle, is unknown, but all its sides are known, then to calculate the length of the bisector we use an additional variable, which we call a semiperimeter and denote by P: P = 1/2 * (MK + KB + MB). After this, we make some changes to the previous formula by which the length of the bisector was determined, namely, in the numerator of the fraction we put the doubled square root of the product of the lengths of the sides adjacent to the angle by the half -perimeter and the quotient, where the length of the third side is subtracted from the half -perimeter. We leave the denominator unchanged. In the form of a formula, this will look like this: KA = 2 * √ (MK * KB * P * (P-MB)) / (MK + KB).

The bisector in a right-angled triangle hasall the same properties as in the ordinary, But, in addition to the already known, there is also a new one: the bisectors of the acute angles of a right-angled triangle form an angle of 45 degrees at the intersection. If necessary, it is easy to prove using the properties of a triangle and adjacent angles.

Биссектриса равнобедренного треугольника вместе с has several properties in common. Let's remember what kind of triangle it is. In such a triangle, the two sides are equal, and the angles adjacent to the base are equal. Hence it follows that bisectors that descend to the lateral sides of an isosceles triangle are equal to each other. In addition, the bisector, lowered to the base, is both a height and a median.