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 trigonometry). And, when you've worked your way though all of this, there are some orthocenter properties waiting for you, and some bonus special cases...
What is the orthocenter of a triangle? Orthocenter definition
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 a side and touches the corner opposite to 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).
- Find the slope of one side of the triangle, e.g. AB. Use the slope formula:
slope = (y₂ - y₁) / (x₂ - x₁)
- 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
- 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)
y = y₃ - (x₂ - x₁) * (x - x₃) / (y₂ - y₁)
- Repeat the steps for another side, either AC or BC.
y = y₂ - (x₃ - x₁) * (x - x₂) / (y₃ - y₁)
- 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).
- Find the slope:
AB side slope = (5 - 1) / (3 - 1) = 2
- Calculate the slope of the perpendicular line:
perpendicular slope to AB side = - 1/2
- Find the line equation:
y - 2 = - 1/2 * (x - 7) so
y = 5.5 - 0.5 * x
- 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
- Solve the system of linear equations:
y = 5.5 - 0.5 * x and
y = -1/3 + 4/3 * x
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.
A more compact formula for find a triangle's orthocenter exists, but you need to be familiar with the concept of the tangent. 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.
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.