The research of a triangle occupied mathematicians throughout centuries. The most part of the properties and theorems connected with triangles uses special lines of a figure: median, bisector and height.
Median and its properties
The median is one of the main lines of a triangle. This piece and a straight line on which it lies connects a point at the head of a triangle corner to the middle of the opposite party of the same figure. In an equilateral triangle the median is also a bisector and height.
The property of a median which will significantly facilitate the solution of many tasks consists in the following: if in a triangle to carry out medians from each corner, then all of them, being crossed in one point, 2:1 will share in the ratio. The ratio should be counted from vertex of angle.
The median has property to divide everything equally. For example, any median divides a triangle into two others, equal on the area. And if to carry out all three medians, then in a big triangle it will turn out 6 small, also equal on the area. Such figures (with an identical area) are called equal.
Bisector
The bisector represents a beam which begins in vertex of angle and halves the same corner. The points lying on this beam are equidistanted from sides of angle. Properties of a bisector well help with the solution of the tasks connected with triangles. In a triangle a bisector call a piece which lies on a beam of a bisector and connects top to the opposite party. The point of intersection with the party divides it into pieces which relation is equal to the relation adjacent to them to the parties. If to enter a circle in a triangle, then its center will coincide with a point of intersection of all bisectors of this triangle. This property has reflection and in stereometry - there a role of a triangle is played by a pyramid, and circles - a sphere.
Height
Also as the median and a bisector, height in a triangle first of all connect vertex of angle and the opposite party. Communication results in the following: height is the perpendicular which is carried out from top to a straight line which comprises the opposite party. If height is carried out in a rectangular triangle, then, concerning the opposite side, it divides all triangle into two others which are in turn similar to the first. Quite often the concept of a perpendicular is applied in stereometry to define interpositions of straight lines in the different planes and distance between them. In this case the piece performing function of a perpendicular has to have a right angle with both straight lines. Then the numerical value of this piece will show distance between two figures.