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In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side.
In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles. [6] The vertex opposite the base is called the apex. [7] In the ...
If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side). The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
In a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, π radians, two right angles, or a half-turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides. It was unknown for a long time whether other geometries exist, for which this sum is different. The influence of this ...
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. [48] Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. [49]
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. [ 1 ] [ 2 ] This statement permits the inclusion of degenerate triangles , but some authors, especially those writing about elementary geometry, will exclude this ...
The large triangle that is inscribed in the circle gets subdivided into three smaller triangles, all of which are isosceles because their upper two sides are radii of the circle. Inside each isosceles triangle the pair of base angles are equal to each other, and are half of 180° minus the apex angle at the circle's center.
The midpoint theorem generalizes to the intercept theorem, where rather than using midpoints, both sides are partitioned in the same ratio. [1] [2] The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle.