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Orthocenter Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Jan 18, 2024


Welcome to the orthocenter calculator – a tool where you can easily find the orthocenter of any triangle, be it right, obtuse, or acute. If you're uncertain what the orthocenter of a triangle is, we've prepared a nice explanation, as well as an orthocenter definition. Afterward, you can learn how to find the orthocenter with a step-by-step set of instructions (or you can just use the orthocenter formula, fueled by the trigonometry calculator). And, when you've worked your way through all of this, there are some orthocenter properties waiting for you and some bonus special cases...

What is the orthocenter of a triangle?

The orthocenter of a triangle is the point where the altitudes of the triangle intersect. The three altitudes of a triangle are always concurrent, meaning that they meet at the same point. As a quick reminder, the altitude is the line segment that is perpendicular to a side and touches the corner opposite the side.

How to find orthocenter?

Now that you've been introduced to the orthocenter definition let's check how to find it. The easiest, most straightforward way to calculate the orthocenter of a triangle is to follow this step-by-step guide:

To start, let's assume that the triangle ABC has the vertex coordinates A = (x₁, y₁), B = (x₂, y₂), and C = (x₃, y₃).

Image of a triangle with vertices and orthocenter marked, Cartesian coordinate system
  1. Find the slope of one side of the triangle, e.g., AB. Use the slope calculator or the below formula:

    slope = (y₂ - y₁) / (x₂ - x₁)

  2. Calculate the slope that is perpendicular to side AB. That way, you'll find the slope of the triangle's altitude for that side. The equation for the altitude's slope is:

    perpendicular slope = -1 / slope

  3. Then, you need to find the equation for the line containing the triangle's altitude – the one that goes through vertex C (x₃, y₃). Use the equation for the point slope formula:

    y - Y = m × (x - X)

    For our example, it will be:

    y - y₃ = m × (x - x₃),

    where m = -1 / slope = - (x₂ - x₁) / (y₂ - y₁)

    therefore:

    y = y₃ - (x₂ - x₁) × (x - x₃) / (y₂ - y₁)

  4. Repeat the steps for another side, either AC or BC.

    y = y₂ - (x₃ - x₁) × (x - x₂) / (y₃ - y₁)

  5. Solve the system of linear equations (two equations in slope intercept form) to find the orthocenter.

How to find orthocenter – an example

The equations in the above paragraph may look scary, but you don't need to worry, it's not that difficult! Let's check how to find the orthocenter with an example where our triangle ABC has the vertex coordinates: A = (1, 1), B = (3, 5), C = (7, 2).

  1. Find the slope:

    AB side slope = (5 - 1) / (3 - 1) = 2

  2. Calculate the slope of the perpendicular line:

    perpendicular slope to AB side = - 1/2

  3. Find the line equation:

    y - 2 = - 1/2 × (x - 7) so y = 5.5 - 0.5 × x

  4. Repeat for another side, e.g., BC;

    BC side slope = (2 - 5) / (7 - 3) = - 3/4

    perpendicular slope to BC side = 4/3

    y - 1 = 4/3 × (x - 1) so y = -1/3 + 4/3 × x

  5. Solve the system of linear equations:

    y = 5.5 - 0.5 × x and
    y = -1/3 + 4/3 × x

    so

    5.5 - 0.5 × x = -1/3 + 4/3 × x

    35/6 = x × 11/6

    x = 35/11 ≈ 3.182.

    Substituting x into either equation will give us:

    y = 43/11 ≈ 3.909

Of course, you'll obtain the same result from our orthocenter calculator💪! Just type the three triangle vertices, and we'll calculate the orthocenter coordinates for you.

Orthocenter formula

Image of a triangle with vertices, angles and orthocenter marked, Cartesian coordinate system

A more compact formula for finding a triangle's orthocenter exists, but you need to be familiar with the concept of the tangent, which we described in the tangent calculator. To find the orthocenter coordinates H = (x, y), you need to solve these equations:

x = (x1 × tan(α) + x2 × tan(β) + x3 × tan(γ)) / (tan(α) + tan(β) + tan(γ))

y = (y1 × tan(α) + y2 × tan(β) + y3 × tan(γ)) / (tan(α) + tan(β) + tan(γ))

While those orthocenter formulas may look way easier than the previous instructions on how to find the coordinates of the center, you probably don't have the triangle's angles, α, β, and γ, provided, do you?

So you'll probably need to find them first. Use the Pythagorean theorem to find the length of the triangle's sides. Then apply the law of cosines to find the angles of the triangle. Our orthocenter calculator has all of this built in, but if you want to do it step by step, then you can use the Pythagorean theorem calculator and law of cosines calculator.

Orthocenter properties and trivia

There are some interesting orthocenter properties! The orthocenter:

  • coincides with the circumcenter, incenter and centroid for an equilateral triangle,
  • coincides with the right-angled vertex for right triangles,
  • lies inside the triangle for acute triangles,
  • lies outside the triangle in obtuse triangles.

Did you know that...

  • three triangle vertices and the triangle orthocenter of those points form the orthocentric system. If you make a triangle out of any three of these points, the remaining one will be its orthocenter.
  • reflection of the orthocenter over any of the three sides lies on the circumcircle of the triangle.
  • the angle formed at the orthocenter is supplementary to the angle at the vertex.
  • in every non-equilateral triangle, there's a line going through all important triangle centers (orthocenter, centroid, circumcenter, nine-point circle) - it's called Euler line.

FAQ

Is orthocenter and circumcenter the same?

No, in general, they are two different points. The orthocenter is where the three altitudes of the triangle meet, while the circumcenter is where the perpendicular bisectors meet. Sometimes, the orthocenter and circumcenter coincide, for example, in the equilateral triangle.

What is the orthocenter of the 3 4 5 triangle?

The 3 4 5 triangle is a right-angled triangle. (To verify this claim, it suffices to plug in these values into the Pythagorean theorem.) The orthocenter of a right-angled triangle coincides with the right-angled vertex. In our case, it is the vertex that joins the sides of lengths 3 and 4.

Is the orthocenter equidistant from the vertices?

In general, no. The point that is equidistant from the vertices is the circumcenter of the triangle. If the orthocenter coincides with the circumcenter, then the orthocenter is equidistant from the vertices. This happens in equilateral triangles.

How do I construct the orthocenter of a triangle with a compass and straightedge?

Assume your triangle has vertices ABC.

  1. Set the compasses' width to the length of AB.
  2. Put the compasses on A and draw an arc across the side BC. Label the point of intersection D.
  3. Put the compasses on B and set their width to more than half the distance between B and D.
  4. From B and D, draw two arcs that intersect, creating point E.
  5. The line through A and E contains the altitude.
  6. Repeat for another side. The altitudes intersect at the orthocenter.
Hanna Pamuła, PhD
Image of a triangle with vertices and orthocenter, Cartesian coordinate system
Vertex A
x₁
y₁
Vertex B
x₂
y₂
Vertex C
x₃
y₃
Orthocenter H
x
y
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