# Orthocenter of a Triangle

How to find the orthocenter of a triangle: definition, example, and its solution.

## Definition

The orthocenter of a triangle

is the intersecting point

of three heights (altitudes) of a triangle.

## Example

To find the orthocenter *M*,

find the intersection of these two heights.

So find the linear equations of these two heights.

First, find the linear equation of the first height.

The endpoints of *BC* are (-3, -1) and (6, -1).

Their *y* values are -1.

So *BC* is a horizontal side.*BC* and the height are perpendicular.

So the first height is a vertical segment.*A*(3, 5) is on the blue height.

So the linear equation of the first height is*x* = 3.

Slope of a line - No slope

Next, find the linear equation of the second height.*C*(6, -1) is on the height.

The slope is not given.

But you can find the slope of the height

by using the slope of *AB*,

because *AB* and the height are perpendicular.

(You'll see the details below.)

So find the slope of *AB*: *m*_{AB}.

Its endpoints are (-3, -1) and (3, 5).

So *m*_{AB} = [5 - (-1)]/[3 - (-3)]

= 1.

Slope of a line

*m*_{AB} = 1

Set the slope of the height *m*.*AB* and the height are perpendicular.

So 1⋅*m* = -1.

So *m* = -1.

Perpendicular lines

The slope of the height, *m*, is -1.

And *C*(6, -1) is on the height.

Then the linear equation in point-slope form is*y* = -(*x* - 6) - 1.

Point-slope form

-(*x* - 6) = -*x* + 6

6 - 1 = 5

So the linear equation of the second height is*y* = -*x* + 5.

The linear equation of the heights are

[*x* = 3] and [*y* = -*x* + 5].

The intersection is the orthocenter *M*.*M* is on [*x* = 3].

So the *x* value of *M* is 3.

To find the *y* value of *M*,

put [*x* = 3] into [*y* = -*x* + 5].

Then *y* = 2.

Substitution method

*x* = 3*y* = 2

So the orthocenter *M* is (3, 2).