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unit of measure: the base unit | the supplementary unit | the derived unit

geometry: the area of a flat figure | the area of a solid | the volume of the solid


A triangle

Definition:

A triangle is a polygon formed by the union of three segments, where the points of intersection are non-collinear points. Points of intersection are called vertexes, and the segments are referred to as sides.

 

The scalene triangle is the triangle which has all sides of different lengths.

The acute-angled triangle is a variety of scalene triangle, which is characterized by having acute angles, all of them measuring less than 90°.

The obtuse triangle is a variety of scalene triangle, which is characterized by having an obtuse angle, that is, measuring more than 90°.

The right triangle is a variety of scalene triangle, which is characterized by having a right angle. The sides adjacent to the right angle are called legs, and the opposite side – hypotenuse.

The Kepler triangle is a variety of right triangle characterized by having a right angle, and the sides constitute a geometrical progression. In the same row, the ratio of sides’ length are related to the golden ratio.

The isosceles triangle is a variety of scalene triangle, which has two congruent sides (equal). The third side is called base.

The equilateral triangle is a variety of scalene and isosceles triangle, which is characterized by the congruence (equality) of all its sides and angles.

Proprietes:

The perimeter is a sum of all its sides.

The mediatrix is the perpendicular line on a segment through its middle. Mediatrices of the three sides of a triangle are called mediatrices of a triangle.

The median is the segment of a line joining a vertex of a line to the midpoint of the opposing side.

The height is a segment joining a vertex to the opposite side or to its extension, forming a right angle.

The bisector is an inner semi-line with its origin in the vertex and divides the angle in two congruent angles. Bisectors of those three inner angles of a triangle are called inner bisectors.

The center of a circumscribed circle of a triangle is the intersection of those three mediatrices (perpendicular on the midpoint of each side) of the given triangle. Center of a circumscribed circle is in the interior of the triangle (in case of acute-angled triangles) or in the exterior of the triangle (in case of obtuse triangle). In case of right triangle, the center of circumscribed circle is on the hypotenuse, on its midpoint.

The center of a circle inscribed in a triangle is on the intersection of those three bisectors of the inner angles of the triangle.

The orthocenter of a triangle is on the intersection of three heights of the given triangle. The orthocenter is in the interior of triangle (in case of acute-angled triangles) or in the exterior of the triangle (in case of obtuse triangle). In case of right triangle, the orthocenter is the vertex of the right angle.

The weight center of a triangle is the intersection of three medians of the triangle. The weight center is on every median at the distance of 2/3 from the vertex and 1/3 from base.

The middle line is the segment determined by the midpoints of two sides of a triangle. It is parallel with the third side and is equal to one half of its length.

The orthocenter, weight center and center of a circumscribed circle of a triangle are collinear. They form the Euler line.

 

context information: contemporary and archaic unit of measurement of area