Altitude of a triangle

Builds on Triangle Line Vector projection Distance from a point to a line

Definition

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$ , and let $\cdot$ be the dot product. Let $H_A$ be the orthogonal projection of $A$ onto the line $BC$ : $H_A = B + \frac{(A - B) \cdot (C - B)}{(C - B) \cdot (C - B)} \, (C - B).$ The altitude of $ABC$ from $A$ is the segment from $A$ to $H_A$ . The point $H_A$ is the foot of the altitude, and $h_a = \|A - H_A\|$ is its length. The altitudes from $B$ and from $C$ are defined in the same way.

Remarks

The formula uses the vector projection of $A - B$ onto $C - B$ . The length $h_a$ is the distance from $A$ to the line $BC$ . The word altitude also names the whole line through $A$ and $H_A$ , or the length $h_a$ . Mathlib's Affine.Simplex.altitude is the line. With $a = \|C - B\|$ , the area of the triangle is $\frac{1}{2} a h_a$ . Let the right angle of a right triangle be at $C$ . Then the leg $AC$ is the altitude from $A$ , and the leg $BC$ is the altitude from $B$ .

Reading path

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Euclidean distance concept
Line concept
Normal vector concept
Triangle concept
Vector projection concept
Distance from a point to a line statement
Altitude of a triangle definition

Tag	`0X89GC65N1`
URI	`m2://0X89GC65N1`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Altitude of a triangle
Uses	Triangle, Line, Vector projection, Distance from a point to a line

Altitude of a triangle

Definition

Remarks

Reading path

Further reading

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