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Circumcenter Theorem

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Circumcenter Theorem

Study Guide

Key Definition

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is equidistant from the vertices of the triangle.

Important Notes

The circumcenter can be inside, outside, or on the triangle, depending on the type of triangle.
In a right triangle, the circumcenter is at the midpoint of the hypotenuse.
The circumcenter of an obtuse triangle is outside the triangle.
The circumcenter of an acute triangle is inside the triangle.

Mathematical Notation

$\overline{AB}$ denotes a line segment from point A to point B.

$\triangle ABC$ represents a triangle with vertices A, B, and C.

$O$ denotes the circumcenter of triangle ABC.

Remember to use proper notation when solving problems. All correctAnswer indices use 0-based numbering.

Why It Works

The circumcenter works because of the property of perpendicular bisectors: any point on a perpendicular bisector is equidistant from the endpoints of the segment. So, the circumcenter is equidistant from the vertices of the triangle.

Remember

The circumcenter is the center of the circle that passes through all the vertices of the triangle, called the circumcircle.

Quick Reference

Circumcenter:Point of intersection of the perpendicular bisectors of a triangle

Understanding Circumcenter Theorem

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is equidistant from all vertices, making it the center of the circumcircle.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Where is the circumcenter located in an obtuse triangle? (Refer to Fig.1)

Real-World Problem

Question Exercise

Intermediate

Mapping a Park

You are designing a park shaped as a triangle with corners at coordinates A(0,0), B(6,0), and C(0,8). You want to place a water fountain equidistant from the three corners. Where should you place the fountain? (Refer to Fig.2)

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove that the circumcenter of a right triangle is at the midpoint of the hypotenuse. (Refer to Fig.3)

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following properties is unique to the circumcenter of a triangle?

Recap

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Review key concepts and takeaways