# Triangle height calculator

## Height

## Triangle

## Triangle height calculator

This triangle height calculator will help you find all three altitudes of a triangle, knowing the coordinates of the vertices, or the length of the sides of the triangle.

The height of a triangle is the perpendicular drawn from the vertex of the triangle to the opposite side of the line containing it.

Depending on the type of triangle, the height can be inside the triangle (for an acute triangle), coincide with its side (for a right triangle - a page), or intersect the outer area of the triangle (for an obtuse triangle).

## Triangle orthocenter properties

The heights of a triangle intersect at one point, which is called the orthocenter. This statement is easy to prove using vector identity for any A, B, C, H points (not necessarily the same). \overrightarrow{EA} * \overrightarrow{BC} + \overrightarrow{EB}*\overrightarrow{CA} + \overrightarrow{EC}*\overrightarrow{AB} = 0

To prove the identity, use \overrightarrow{AB} = \overrightarrow{HB} - \overrightarrow{HA}, \ \overrightarrow{BC} = \overrightarrow{HC} - \overrightarrow{HB}, \ \overrightarrow{CA} = \overrightarrow{HA} - \overrightarrow{HC} formulas. For point H, you need to take the intersection point of the two heights of the triangle.

## Equilateral Triangle Elevation Properties

- If two heights of a triangle are equal, then the triangle is equilateral (the Steiner-Lemus theorem), and the third height is both the midpoint of the angle and the midline from which it exits.
- Two heights of an equilateral triangle are equal, and the third height is at the same time the median and the bisector.
- All three heights of an equilateral triangle are equal.

## Basic Formulas

- Height of an isosceles triangle, through a side and an angle: h_a = b*sin(\gamma) = c*sin(\beta)
- Height of a triangle in terms of area: h_a = \dfrac{2S}{a} where S - triangle area, a - the length of the side to which the height is attached.
- Triangle height through the radius of the circumscribed circle: h_a = \dfrac{bc}{2R} where bc - a product of lateral edges, R - radius of the circumscribed circle.
- Triangle height through the radius of the inscribed circle: \dfrac{1}{h_a} + \dfrac{1}{h_b} + \dfrac{1}{h_c} = \dfrac{1}{r} where r - radius of the inscribed circle.
- The height of the triangle through the semi-perimeter: h_a = \dfrac{2\sqrt{p(p-a)(p-b)(p-c)}}{a} where a - side on which the height falls, b, c - the lengths of the other two sides, p - semi-perimeter .
- Height of an equilateral triangle: h = a\dfrac{\sqrt{3}}{2} where a - side of a triangle.

## Tags

- Area
- Volume
- Perimeter
- Side
- Height
- Diagonal
- Radius
- Median
- Bisector
- Angle
- Theorems