Triangle Centers
A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.
Every triangle has four classical centers. The centroid is where the three medians meet (each median connects a vertex to the midpoint of the opposite side). The incenter is where the three angle bisectors meet (it is the center of the inscribed circle). The circumcenter is where the three perpendicular bisectors meet (it is the center of the circumscribed circle). The orthocenter is where the three altitudes meet.
What you'll learn
- Identify the centroid as the intersection of medians
- Identify the incenter as the intersection of angle bisectors
- Identify the circumcenter as the intersection of perpendicular bisectors
- Identify the orthocenter as the intersection of altitudes
Worked example
Problem. What is the centroid of a triangle?
- A median connects a vertex to the midpoint of the opposite side.
- The three medians always meet at one point — the centroid.
- The centroid is also the triangle's center of mass.
- Connect the result back to Triangle Centers so the geometric relationship is explicit.
Answer: The intersection of the three medians.
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. Which center is the intersection of the three medians?
Choices: Centroid · Incenter · Circumcenter · Orthocenter
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- Medians connect each vertex to the opposite side's midpoint.
Answer: Centroid
2. Which center is the intersection of the three angle bisectors?
Choices: Centroid · Incenter · Circumcenter · Orthocenter
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- Angle bisectors meet at the center of the inscribed circle.
Answer: Incenter
3. Which center is the intersection of the three perpendicular bisectors?
Choices: Centroid · Incenter · Circumcenter · Orthocenter
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- Perpendicular bisectors meet at the center of the circumscribed circle.
Answer: Circumcenter
4. Which center is the intersection of the three altitudes?
Choices: Centroid · Incenter · Circumcenter · Orthocenter
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- Altitudes are perpendicular from a vertex to the opposite side.
Answer: Orthocenter
5. The centroid is also the triangle's:
Choices: Center of mass · Center of the inscribed circle · Center of the circumscribed circle
Show solution
- Balance any triangular shape at its centroid and it stays level.
Answer: Center of mass
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