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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

Why it matters: Triangle centers appear in structural engineering (centroid as center of mass), surveying (circumcenter equidistant from three landmarks), and design (incenter as inscribed-circle center for a maximum inner space).

Worked example

Problem. What is the centroid of a triangle?

  1. A median connects a vertex to the midpoint of the opposite side.
  2. The three medians always meet at one point — the centroid.
  3. The centroid is also the triangle's center of mass.

Answer: The intersection of the three medians.

Practice problems

1. Which center is the intersection of the three medians?

Choices: Centroid · Incenter · Circumcenter · Orthocenter

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which center is the intersection of the three medians?
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. Medians connect each vertex to the opposite side's midpoint.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Centroid

2. Which center is the intersection of the three angle bisectors?

Choices: Centroid · Incenter · Circumcenter · Orthocenter

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which center is the intersection of the three angle bisectors?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Angle bisectors meet at the center of the inscribed circle.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Incenter

3. Which center is the intersection of the three perpendicular bisectors?

Choices: Centroid · Incenter · Circumcenter · Orthocenter

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which center is the intersection of the three perpendicular bisectors?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Perpendicular bisectors meet at the center of the circumscribed circle.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Circumcenter

4. Which center is the intersection of the three altitudes?

Choices: Centroid · Incenter · Circumcenter · Orthocenter

Show solution
  1. Core Practice: First identify exactly what the question is asking: Which center is the intersection of the three altitudes?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Altitudes are perpendicular from a vertex to the opposite side.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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
  1. Core Practice: First identify exactly what the question is asking: The centroid is also the triangle's:
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Balance any triangular shape at its centroid and it stays level.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Center of mass

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