Triangle - Further Formulas For General Euclidean Triangles

Further Formulas For General Euclidean Triangles

The formulas in this section are true for all Euclidean triangles.

The medians and the sides are related by

and

,

and equivalently for m_b and m_c.

For angle α opposite side a, the length of the internal bisector is given by

for semiperimeter s, where the bisector length is measured from the vertex to where it meets the opposite side.

The following formulas involve the circumradius R and the inradius r:

where h_a etc. are the altitudes to the subscripted sides;

and

.

Suppose two adjacent but non-overlapping triangles share the same side of length f and share the same circumcircle, so that the side of length f is a chord of the circumcircle and the triangles have side lengths (a, b, f) and (c, d, f), with the two triangles together forming a cyclic quadrilateral with side lengths in sequence (a, b, c, d). Then

Let M be the centroid of a triangle with vertices A, B, and C, and let P be any interior point. Then the distances between the points are related by

Let p_a, p_b, and p_c be the distances from the centroid to the sides of lengths a, b, and c. Then

and

The product of two sides of a triangle equals the altitude to the third side times the diameter of the circumcircle.

Carnot's Theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius. Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle. This method is especially useful for deducing the properties of more abstract forms of triangles, such as the ones induced by Lie algebras, that otherwise have the same properties as usual triangles.

Euler's theorem states that the distance d between the circumcenter and the incenter is given by

or equivalently

where R is the circumradius and r is the inradius. Thus for all triangles R ≥ 2r, with equality holding for equilateral triangles.

If we denote that the orthocenter divides one altitude into segments of lengths u and v, another altitude into segment lengths w and x, and the third altitude into segment lengths y and z, then uv = wx = yz.

The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter.

The sum of the squares of the distances from the vertices to the orthocenter plus the sum of the squares of the sides equals twelve times the square of the circumradius.