Orthocenter of a Triangle

Orthocenter of a Triangle

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

Definition

The orthocenter of a triangle is the intersecting point of three heights (altitudes) of a triangle.

The orthocenter of a triangle is the intersecting point
of three heights (altitudes) of a triangle.

Example

Find the coordinates of the centriod of triangle ABC. A(3, 5), B(-3, -1), C(6, -1).

To find the orthocenter,
find the intersecting point of two heights (purple, blue).

The purple height is perpendicular to BC,
which is horizontal.

So the purple height is a vertical segment.
And it passes through A(3, 5).

So the purple height's linear equation is x = 3.

To find the linear equation of the blue height,
you need to find the slope of the blue height.

And to find the slope of the blue height,
you need to find the slope of AB: mAB

mAB = 1.

The blue height is perpendicular to AB.

So (blue height's slope, m)⋅(mAB, 1) = -1

Perpendicular lines

You got the blue height's slope: m = -1.

And the blue height passes through C(6, -1).

So the blue height's linear equation is
y - (-1) = -(x - 6).

Point-slope form

The blue height's linear equation is y = -x + 5.

The orthocenter is the intersecting point of
x = 3 (purple height) and y = -x + 5 (blue height).

So, by solving this system,
y = 2.

Substitution method

So M(3, 2).