Calculating the Circumcenter of a Triangle | Steps and Formulas

circumcenter of a triangle

The circumcenter of a triangle is the point where the perpendicular bisectors of the triangle’s sides intersect

The circumcenter of a triangle is the point where the perpendicular bisectors of the triangle’s sides intersect. In simpler terms, it is the center of the circle that passes through all three vertices of the triangle.

To find the circumcenter, you need to follow these steps:

1. Take the three given vertices of the triangle: A (x1, y1), B (x2, y2), and C (x3, y3).

2. Calculate the midpoints of two of the sides of the triangle. Let’s take side AB as an example. The midpoint MAB can be found using the midpoint formula:

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

3. Find the slope of the line passing through points A and B. The slope formula can be used:

Slope of AB = (y2 – y1)/(x2 – x1)

4. Calculate the negative reciprocal of the slope of AB. Let’s call it MAB⊥.

MAB⊥ = -1 / (Slope of AB)

5. Now, we can find the equation of the perpendicular bisector of side AB. The equation can be obtained using the point-slope form of a line:

Perpendicular bisector of AB: y – yMAB = MAB⊥ (x – xMAB)

6. Repeat steps 2-5 for another pair of sides, such as BC.

7. Solve the equations of the two perpendicular bisectors obtained in steps 5 and 6 simultaneously. This can be done by setting the two equations equal to each other and solving for x and y.

8. The resulting x and y coordinates are the coordinates of the circumcenter of the triangle.

It’s important to note that the circumcenter is not always inside the triangle. If the triangle is obtuse, the circumcenter will be outside the triangle. If the triangle is right-angled, the circumcenter will be at the midpoint of the hypotenuse.

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