The angle bisector and its properties

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2018-03-24 23:27:08

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Among the numerous items of secondary schools is the same as “geometry”. Traditionally it is believed that the ancestors of this systematic science are Greeks. To date, the Greek geometry, called elementary, as it is the beginning of the study of the simplest forms: planes, straight lines, regular polygons and triangles. At last we stop their attention, and more precisely on the bisector of this figure. For those who have forgotten, the angle bisector is a segment bisector of one of the angles of a triangle that divides it in half and connects the top with the point placed on the opposite side.

The angle Bisector has a number of properties that you need to know when solving certain tasks:

  • Bisector of an angle is a geometrical place of points that were deleted at equal distances from adjacent to the corner sides.
  • Bisector in a triangle divides the opposite angle side into segments that are proportional to the adjacent sides. For example, given a triangle MKB, where the angle K is out of the bisector connecting the top of the corner point A on the opposite side of the MB. After analyzing this property and our triangle have the MA/AB=MK/KB.
  • The Point at which the bisectors intersect all three angles of a triangle is the center of the circle, which is inscribed in the same triangle.
  • The Basis of 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 the two bisectors of one triangle are equal, then the triangle is isosceles.

It Should be noted that if there are three bisectors, construct a triangle on them, even with a compass it is impossible.

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Very often when solving problems bisector of a triangle is unknown, but it is necessary to determine its length. To solve this problem it is necessary to know the angle that the bisector is divided in half, and adjacent to that corner part. In this case the required length is defined as the ratio of twice the work adjacent to the corner sides and the cosine of the angle divided by half the sum of adjacent to the corner sides. For example, given the same triangle MKB. The bisector out of the corner of K and intersects the opposite direction MV at the point A. the Angle from where the bisector, denoted by y. Now write down everything that is said with words in a formula: KA = (2*MK*KB*cos y/2) / ( MK+KB).

If the value of the angle from where the bisector of the triangle is unknown, but known to all parties, to calculate the length of the bisector we will use an additional variable, which we'll call properiter and denote by the letter P : P=1/2*(MK+KB+MB). Then make some changes in the previous formula, which determined the length of the bisector, namely to the numerator of the fraction put twice the square root of the product of the lengths of the sides adjacent to the corner on properiter and private, where properiety is subtracted from the length of the third side. The denominator stay the same. In a formula it would look like this: KA=2*√(MK*KB*P*(P-MB)) / ( MK+KB).

The Bisector of a right triangle has all the same properties as normal, But, in addition to the already known, is and new: the bisectors of the acute angles of a right triangle with the intersection form an angle of 45 degrees. If necessary, it is easy to prove using properties of a triangle and supplementary angles.

Bisector of an isosceles triangle with common properties has some of my own. Remember that this is for the triangle. Such a triangle two sides are equal, and equal adjacent to the base corners. It follows that the bisectors, which descend to the sides of an isosceles triangle are equal. In addition, bisector, is lowered onto the base is also the altitude and median.

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