Circumcenter Calculator

What is the Circumcenter of a Triangle?

The circumcenter of a triangle is a special point where all three perpendicular bisectors of the triangle's sides meet. It's like finding the center of a circle that touches all three corners of the triangle!

How to Calculate the Circumcenter of a Triangle

To find the circumcenter, we use the coordinates of the triangle's three corners. It's like solving a puzzle to find the perfect spot that's equally far from all three points.

Formula

The magic formula for finding the circumcenter of a triangle with corners (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

\[ U_x = \frac{(x_1^2 + y_1^2)(y_2 - y_3) + (x_2^2 + y_2^2)(y_3 - y_1) + (x_3^2 + y_3^2)(y_1 - y_2)}{2[(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2))]} \]

\[ U_y = \frac{(x_1^2 + y_1^2)(x_3 - x_2) + (x_2^2 + y_2^2)(x_1 - x_3) + (x_3^2 + y_3^2)(x_2 - x_1)}{2[(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2))]} \]

Where:

• \(U_x\) is the x-coordinate of the circumcenter
• \(U_y\) is the y-coordinate of the circumcenter
• \(x_1, x_2, x_3\) are the x-coordinates of the triangle's corners
• \(y_1, y_2, y_3\) are the y-coordinates of the triangle's corners

Calculation Steps

1. Write down the coordinates of your triangle's corners
2. Calculate the denominator: \(D = 2[(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2))]\)
3. Calculate \(U_x\) using the formula above
4. Calculate \(U_y\) using the formula above
5. The results (U_x, U_y) are your circumcenter's coordinates!

Example and Visual Representation

Let's find the circumcenter of a triangle with corners at (0, 0), (4, 0), and (2, 3):

Step 1: We have (x₁, y₁) = (0, 0), (x₂, y₂) = (4, 0), (x₃, y₃) = (2, 3)

Step 2: \(D = 2[(0(0 - 3) + 4(3 - 0) + 2(0 - 0))] = 24\)

Step 3: \(U_x = \frac{(0^2 + 0^2)(0 - 3) + (4^2 + 0^2)(3 - 0) + (2^2 + 3^2)(0 - 0)}{24} = 2\)

Step 4: \(U_y = \frac{(0^2 + 0^2)(2 - 4) + (4^2 + 0^2)(0 - 2) + (2^2 + 3^2)(4 - 0)}{24} = 1.5\)

Step 5: The circumcenter is at (2, 1.5)

In this picture, you can see our blue triangle. The red dot is the circumcenter, where all three dashed lines (perpendicular bisectors) meet. The purple dashed circle is the circumcircle, which passes through all three corners of the triangle. Isn't it amazing how the circumcenter is exactly in the middle of this special circle?