How to Find the Circumcenter of a Triangle Using the Section Formula

The circumcenter of a triangle is a significant geometric point that represents the center of the circumcircle, a circle that passes through all three vertices of the triangle. This point is where the perpendicular bisectors of the triangle's sides intersect. In this guide, we will explore step-by-step how to find the circumcenter of a triangle using the section formula.

What is the Circumcenter of a Triangle?

The circumcenter of a triangle is the point where the perpendicular bisectors of its sides intersect. It is also the center of the circle that circumscribes the triangle, meaning the circle passes through all three vertices of the triangle.

Determining the Vertices of the Triangle

To find the circumcenter, we first need to identify the vertices of the triangle. Let's denote the vertices as A(x1, y1), B(x2, y2), and C(x3, y3).

Step 1: Find the Midpoints of the Sides

The next step is to calculate the midpoints of at least two sides of the triangle. The midpoints are essential for determining the perpendicular bisectors later on.

Midpoint of AB (denoted as MAB): Midpoint of AC (denoted as MAC):

Using the formula for the midpoint:

MAB ( (x1 x2) / 2, (y1 y2) / 2 )

MAC ( (x1 x3) / 2, (y1 y3) / 2 )

Step 2: Calculate the Slopes of the Sides

We now calculate the slopes of the sides of the triangle.

Slope of AB (denoted as mAB): Slope of AC (denoted as mAC):

The slope of a line is given by the formula:

mAB (y2 - y1) / (x2 - x1)

mAC (y3 - y1) / (x3 - x1)

Step 3: Determine the Slopes of the Perpendicular Bisectors

Since the perpendicular bisectors are perpendicular to the sides of the triangle, they have slopes that are the negative reciprocals of the slopes of the sides.

Slope of the perpendicular bisector of AB (denoted as mPBAB)): Slope of the perpendicular bisector of AC (denoted as mPBAC)):

The formula for the slope of a perpendicular bisector is:

mPBAB -1 / mAB

mPBAC -1 / mAC

Step 4: Write the Equations of the Perpendicular Bisectors

Having the slopes, we can now write the equations of the perpendicular bisectors using the point-slope form of the equation of a line.

For AB:
In point-slope form, y - y1' mPBAB (x - x1'')
Substitute the midpoint of AB, then:
y - (y1 y2) / 2 ( -1 / (y2 - y1) ) (x - (x1 x2) / 2) For AC:
In point-slope form, y - y1' mPBAC (x - x1'')
Substitute the midpoint of AC, then:
y - (y1 y3) / 2 ( -1 / (y3 - y1) ) (x - (x1 x3) / 2)

Step 5: Solve the System of Equations

Finally, by solving the system of equations for the perpendicular bisectors, we can find the coordinates of the circumcenter O(x, y).

An Example

Let's consider a triangle with vertices A(0, 0), B(4, 0), and C(2, 3).

Midpoints: MAB ( (0 4) / 2, (0 0) / 2 ) (2, 0) MAC ( (0 2) / 2, (0 3) / 2 ) (1, 1.5) Slopes of the Sides: mAB (0 - 0) / (4 - 0) 0 (vertical line) mAC (3 - 0) / (2 - 0) 1.5 Slopes of the Perpendicular Bisectors: mPBAB undefined (vertical line x 2) mPBAC -1 / 1.5 -2 / 3

The equation for the perpendicular bisector of AB is x 2, and the equation for the perpendicular bisector of AC can be found using the point-slope form.

Solving the system of equations gives us the coordinates of the circumcenter O.

Conclusion

The method to find the circumcenter involves the intersection of the perpendicular bisectors. This geometric and algebraic approach makes the calculation both systematic and efficient.