Centers of a triangle
High School
Definition
Triangles can have thousands of different centers. The four main ones are Centroid, Circumcenter, Incenter and Orthocenter. A centroid
Worked examples
\(\)Centroid \( G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\)
The centroid is the average of the three vertices — the center of mass, where medians meet.
\(\)Circumcenter \( O\) is equidistant from all three vertices: \(OA = OB = OC\)
The circumcenter lies where perpendicular bisectors meet; it's the center of the circumscribed circle.
\(\)Incenter \( I\) is equidistant from all three sides
The incenter is where angle bisectors meet — the center of the inscribed circle touching each side.
Common mistakes
- All four centers are always inside the triangle → Circumcenter and orthocenter can lie outside for obtuse triangles Only the centroid and incenter always stay inside any triangle.
- The centroid is the midpoint of one side → The centroid is \(\frac{2}{3}\) along each median from vertex to opposite midpoint Midpoints of sides are not centers; the centroid divides medians in a 2:1 ratio.
- All centers coincide for any triangle → The four centers coincide only for an equilateral triangle In general triangles the centers are distinct points with different properties.
Where you'll use it next
You'll use triangle centers when solving coordinate geometry problems, proving concurrency theorems, and in physics for center-of-mass calculations. They appear again in advanced geometry and engineering applications.
Found in 1 StudyPug lesson
Perpendicular Bisectors: Mastering Formulas and Applications
7th Grade7thGrade 7 Math
Unlock the power of perpendicular bisectors in geometry! Learn formulas, construction techniques, and real-world applications. Enhance your problem-solving skills and ace your exams with our comprehensive guide.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026