Centers of a Triangle

Centers of a Triangle

The main centers of a triangle are listed in the table below along with selected properties.

Table showing 4 triangle centers: circumcenter, incenter, centroid, orthocenter with their concurrent lines and inside/outside...

Key Formula

G = \left(\frac{x_1 + x_2 + x_3}{3},\; \frac{y_1 + y_2 + y_3}{3}\right)

Where:

$G$ = The centroid of the triangle
$(x_1, y_1)$ = Coordinates of vertex A
$(x_2, y_2)$ = Coordinates of vertex B
$(x_3, y_3)$ = Coordinates of vertex C

Worked Example

Problem: Find the centroid of the triangle with vertices A(0, 0), B(6, 0), and C(3, 9).

Step 1: Write down the centroid formula. The centroid is the average of the three vertices' coordinates.

G = \left(\frac{x_1 + x_2 + x_3}{3},\; \frac{y_1 + y_2 + y_3}{3}\right)

Step 2: Substitute the coordinates of A(0, 0), B(6, 0), and C(3, 9).

G = \left(\frac{0 + 6 + 3}{3},\; \frac{0 + 0 + 9}{3}\right)

Step 3: Compute each coordinate.

G = \left(\frac{9}{3},\; \frac{9}{3}\right) = (3,\; 3)

Step 4: Verify: the centroid lies inside the triangle and divides each median in a 2:1 ratio from vertex to midpoint. The midpoint of BC is (4.5, 4.5). The point two-thirds of the way from A(0,0) toward (4.5, 4.5) is (3, 3). ✓

Answer: The centroid is G(3, 3).

Another Example

This example finds the circumcenter instead of the centroid, showing how a different center uses a different construction (perpendicular bisectors rather than averaging coordinates). Note that in a right triangle, the circumcenter lies at the midpoint of the hypotenuse — and indeed the midpoint of BC = ((8+0)/2, (0+6)/2) = (4, 3).

Problem: Find the circumcenter of the triangle with vertices A(0, 0), B(8, 0), and C(0, 6).

Step 1: The circumcenter is the intersection of the perpendicular bisectors of the sides. Start with side AB from (0, 0) to (8, 0). Its midpoint is (4, 0), and since AB is horizontal, its perpendicular bisector is the vertical line x = 4.

x = 4

Step 2: Now find the perpendicular bisector of side AC from (0, 0) to (0, 6). Its midpoint is (0, 3), and since AC is vertical, its perpendicular bisector is the horizontal line y = 3.

y = 3

Step 3: The circumcenter is at the intersection of these two lines.

O = (4,\; 3)

Step 4: Verify by checking that O is equidistant from all three vertices. Distance from O to A: √(16 + 9) = 5. Distance from O to B: √(4 + 9) = √13 ... Let me recheck. OB = √((8−4)² + (0−3)²) = √(16+9) = 5. OC = √((0−4)² + (6−3)²) = √(16+9) = 5. All distances equal 5. ✓

OA = OB = OC = 5

Answer: The circumcenter is O(4, 3), and the circumradius is 5.

Frequently Asked Questions

What are the four main centers of a triangle?

The four classical triangle centers are the centroid (intersection of the three medians), the incenter (intersection of the three angle bisectors), the circumcenter (intersection of the three perpendicular bisectors), and the orthocenter (intersection of the three altitudes). Each always exists for any triangle, but only the centroid and incenter are guaranteed to lie inside the triangle.

Which center of a triangle is always inside the triangle?

The centroid and the incenter always lie inside the triangle, regardless of whether the triangle is acute, right, or obtuse. The circumcenter lies outside an obtuse triangle and on the hypotenuse of a right triangle. The orthocenter lies outside an obtuse triangle and at the vertex of the right angle in a right triangle.

What is the Euler line of a triangle?

The Euler line is a line that passes through the circumcenter, centroid, and orthocenter of any non-equilateral triangle. Remarkably, the centroid always lies exactly one-third of the way from the circumcenter to the orthocenter. In an equilateral triangle, all three points coincide, so no unique line is defined.

Centroid vs. Circumcenter

	Centroid	Circumcenter
Defined by	Intersection of the three medians	Intersection of the three perpendicular bisectors
Formula (coordinates)	Average of the three vertices: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)	Equidistant from all three vertices (solve system of equations)
Always inside?	Yes, always inside the triangle	No — outside for obtuse triangles, on hypotenuse for right triangles
Key property	Center of mass; divides each median in a 2:1 ratio	Center of the circumscribed circle (circumcircle)
On the Euler line?	Yes	Yes

Why It Matters

Triangle centers appear throughout geometry courses, from basic constructions to coordinate geometry proofs. Understanding which center to use matters in practical contexts: the circumcenter locates a point equidistant from three positions (useful in navigation and engineering), while the centroid identifies the balance point of a triangular region. Many competition and standardized test problems require you to distinguish among these centers and apply their properties correctly.

Common Mistakes

Mistake: Confusing the circumcenter with the centroid. Students often assume the centroid is equidistant from all three vertices, or that the circumcenter is the 'average' of the vertices.

Correction: The centroid is the average of the vertex coordinates and serves as the center of mass. The circumcenter is equidistant from all three vertices and is found by intersecting perpendicular bisectors — these are different points with different constructions.

Mistake: Assuming all four centers lie inside the triangle.

Correction: Only the centroid and incenter are always inside. For obtuse triangles, the circumcenter and orthocenter lie outside the triangle. For right triangles, the circumcenter is on the hypotenuse and the orthocenter is at the right-angle vertex.

Related Terms

Triangle — The shape whose centers are studied
Circumcenter — Center of the circumscribed circle
Circumcircle — Circle passing through all three vertices
Incenter — Center of the inscribed circle
Inscribed Circle — Circle tangent to all three sides
Centroid — Intersection of the three medians
Orthocenter — Intersection of the three altitudes
Median of a Triangle — Segment from vertex to opposite midpoint
Altitude of a Triangle — Perpendicular segment from vertex to opposite side