# Triangle: Orthocenter

How to find the orthocenter of a triangle: definition, 1 example, and its solution.

## Definition

### Definition

The orthocenter of a triangle

is the intersecting point

of three heights (altitudes) of a triangle.

## Example

### Example

### Solution

The orthocenter of a triangle

can be found by

finding the intersecting point

of these two heights.

So, find the linear equations

that show these two heights.

First, find this height.

The y values of B and C are both -1.

So BC is a horizontal side.

BC and the height is perpendicular.

So the height is vertical.

The x value of A is 3.

So the linear equation that shows the height is

x = 3.

Next, find the other height.

The linear equation that shows the height

passes through C(6, -1).

But you don't know the slope.

The height is perpendicular to AB.

So, to find the slope of the height,

find the slope of AB.

A(3, 5)

B(-3, -1)

Then the slope of AB is

m_{AB} = [5 - (-1)]/[3 - (-3)].

5 - (-1) = 5 + 1

3 - (-3) = 3 + 3

5 + 1 = 6

3 + 3 = 6

6/6 = 1

So the slope of AB is

m_{AB} = 1.

Set the slope of the height m.

m_{AB} = 1

The height and AB are perpendicular.

So m⋅1 = -1.

Perpendicular Line Equation

So m = -1.

So the slope of the height is

m = -1.

m = -1

The height passes through C(6, -1).

Then the linear equation in point-slope form is

y = -1(x - 6) - 1.

-1(x - 6) = -x + 6

+6 - 1 = +5

So y = -x + 5.

So the linear equation

that shows the second height is

y = -x + 5.

The linear equations that show the slopes are

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

Find the intersecting point,

which is the orthocenter,

by solving this system of linear equations.

Put x = 3

into y = -x + 5.

Then y = -3 + 5.

Substitution Method

-3 + 5 = 2

x = 3

y = 2

So the intersecting point is

M(3, 2).

So the orthocenter is M(3, 2).