Orthocenter Calculator — Find the Orthocenter of Any Triangle

Orthocenter Calculator — Find the Orthocenter of Any Triangle

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 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.

More formally: given a triangle ABC, the three altitudes — from vertex A perpendicular to side BC, from vertex B perpendicular to side AC, and from vertex C perpendicular to side AB — all pass through one common point. That point is the orthocenter, usually denoted H.

How to find the orthocenter?

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).

1. Find the slope of one side of the triangle, e.g., AB:

   slope_AB = (y₂ − y₁) / (x₂ − x₁)

2. Calculate the slope of the perpendicular altitude from vertex C to side AB:

   perpendicular slope = −1 / slope_AB

3. Find the altitude equation through C (x₃, y₃):

   y − y₃ = m × (x − x₃), where m = −(x₂ − x₁) / (y₂ − y₁)

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

   y − y₂ = m' × (x − x₂), where m' = −(x₃ − x₁) / (y₃ − y₁)

5. Solve the system of two linear equations to find the orthocenter coordinates H = (x, y).

How to find the orthocenter — an example

Let's check how to find the orthocenter with an example where our triangle ABC has vertex coordinates: A = (1, 1), B = (3, 5), C = (7, 2).

1. Side AB slope: (5 − 1) / (3 − 1) = 2
   Perpendicular slope from C: −1/2
   Altitude from C: y − 2 = −1/2 × (x − 7) → y = 5.5 − 0.5x
2. Side BC slope: (2 − 5) / (7 − 3) = −3/4
   Perpendicular slope from A: 4/3
   Altitude from A: y − 1 = 4/3 × (x − 1) → y = −1/3 + 4x/3
3. Solve the system:
   5.5 − 0.5x = −1/3 + 4x/3
   35/6 = x × 11/6
   x = 35/11 ≈ 3.182
   y = 43/11 ≈ 3.909

You can verify this result using our orthocenter calculator above — just type the three triangle vertices and you'll get the same answer instantly.

Orthocenter formula

A more compact formula for finding a triangle's orthocenter uses the tangent of the triangle's angles. If the triangle's angles at vertices A, B, C are α, β, γ respectively, then:

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 look compact, you first need the angles α, β, γ. You can find them using the law of cosines applied to the side lengths of the triangle. Our orthocenter calculator has all of this built in automatically.

Orthocenter properties and trivia

Here are some interesting properties of the orthocenter:

• Acute triangle: The orthocenter lies inside the triangle.
• Right triangle: The orthocenter lies exactly at the right-angle vertex.
• Obtuse triangle: The orthocenter lies outside the triangle, beyond the obtuse vertex.
• Equilateral triangle: The orthocenter, circumcenter, centroid, and incenter all coincide at the same central point.
• Euler line: The orthocenter (H), centroid (G), and circumcenter (O) are always collinear. This line is called the Euler line. Moreover, G divides HO in the ratio 2:1.
• Reflection property: Reflecting the orthocenter across the midpoint of any side produces the circumcenter of the triangle.
• Nine-point circle: The midpoints of the segments from the orthocenter to each vertex lie on the nine-point circle of the triangle.

FAQs

Can the orthocenter be outside the triangle?

Yes! For obtuse triangles, the orthocenter always lies outside the triangle. For acute triangles it is inside, and for right triangles it sits exactly on the vertex of the right angle.

What is the difference between the orthocenter and the centroid?

The centroid is the center of mass of the triangle — it is the point where the three medians (lines from each vertex to the midpoint of the opposite side) intersect. The orthocenter is where the three altitudes (perpendicular lines from each vertex to the opposite side) meet. They are different points except in the case of an equilateral triangle.

What is the difference between the orthocenter and the circumcenter?

The circumcenter is equidistant from all three vertices of the triangle — it is the center of the circumscribed circle. The orthocenter is the intersection of the altitudes. Both are related through the Euler line.

How do I use this orthocenter calculator?

Simply enter the X and Y coordinates of all three vertices (A, B, C) of your triangle, optionally select a unit of measurement, and click "Find Orthocenter." The calculator will instantly display the orthocenter coordinates H, the triangle type, side lengths, angles, and more.