Triangle Orthocenter Calculator

What is the Orthocenter of a Triangle?

The orthocenter of a triangle is the point where all three altitudes of the triangle intersect. An altitude is a line segment from a vertex perpendicular to the opposite side (or its extension).

How to Calculate the Orthocenter of a Triangle

To find the orthocenter, follow these steps:

1. Calculate the slopes of two sides of the triangle
2. Find the perpendicular slopes to these sides
3. Determine the equations of the altitude lines
4. Solve the system of equations to find the intersection point (orthocenter)

Calculation Steps

1. Write down the coordinates of your triangle's vertices
2. Calculate the slopes of two sides and their perpendicular slopes
3. Form the equations of two altitude lines
4. Solve the system of equations to find the orthocenter coordinates

Example and Visual Representation

Let's find the orthocenter of a triangle with vertices at A(1, 1), B(3, 5), and C(7, 2):

Step 1: Find the slope of side AB

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

Step 2: Calculate the perpendicular slope to AB

Perpendicular slope to AB = -1/2

Step 3: Find the equation of the altitude from C

y - 2 = -1/2 × (x - 7)

y = 5.5 - 0.5x

Step 4: Repeat for side BC

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

Perpendicular slope to BC = 4/3

y - 1 = 4/3 × (x - 1)

y = -1/3 + 4/3x

Step 5: Solve the system of equations

5.5 - 0.5x = -1/3 + 4/3x

35/6 = 11/6x

x = 35/11 ≈ 3.182

Step 6: Find y-coordinate

y = 5.5 - 0.5 × (35/11) = 43/11 ≈ 3.909

Therefore, the orthocenter is approximately at (3.182, 3.909).

In this diagram, the blue triangle represents ABC, the red dot is the orthocenter O, and the dashed gray lines are the altitudes intersecting at the orthocenter.