created at February 17, 2022

Triangle calculation by given edges

Edges are marked with small Latin letters a, b, c. The opposite angles are marked with Greek letters α, β, γ and according vertices with Capital Latin letters A, B, C.

Perimeter is a sum of all edges:

P = a + b + c

The semiperimeter:

p = \frac{a + b + c}{2}

The area (Heron's formula):

S = \sqrt{p(p - a)(p - b)(p - c)}

Circumscribed and inscribed circles

The circumscribed circle goes through all vertices of the triangle with the center located in the intersection point of edge perpendicular bisectors. The radius is:

R = \frac{a\cdot b\cdot c}{4\cdot S}

The inscribed circle lies inside the triangle and is touching all three edges. The center is located in triangle angle bisector intersection point. The radius is:

r = \frac{S}{p}

Medians

The median is the line connecting the vertex with the opposite edge middle point. They all intersect in one single point, which is the center of mass for the given triangle. The median for the edge c can be found as:

m_c = \sqrt{\frac{2\cdot a^2 + 2\cdot b^2 - c^2}{4}}

Altitudes

The altitude is a line through the vertex and orthogonal to the opposite edge. In other words it is a smallest distance between the vertex and the opposite edge. All the three altitudes intersect in a single point called orthocenter. The altitude to the edge c is:

h_c = \frac{a\cdot b}{2\cdot R} = \frac{2\cdot S}{c}

Angle Bisectors

Angle bisector is a line through the vertex to the opposite edge, which splits the vertex angle into two equal angles. All three bisectors intersect in a single point called incenter. The bisector for the edge c is:

l_c = \frac{2\sqrt{a\cdot b\cdot p\cdot (p - c)}}{a + b}

Angles

Angles can be derived by using Sinus Theorem:

2\cdot R = \frac{a}{sin(\alpha)} = \frac{b}{sin(\beta)} = \frac{c}{sin(\gamma)}

For example, the angle α is:

\alpha = arcsin\left ( \frac{a}{2\cdot R} \right )