5.4 Centers of triangles

Properties of Triangle Centers

Identify and define the incenter, circumcenter, centroid, and orthocenter of a triangle

Each triangle center is defined by a specific type of segment. To find any of these centers, you only need to construct two of the relevant segments; their intersection gives you the point. (The third segment will always pass through the same point, by theorem.)

Incenter

The incenter is where all three angle bisectors meet. Because it lies on every angle bisector, it's equidistant from all three sides of the triangle. That equal distance becomes the radius of the triangle's inscribed circle (incircle), the largest circle that fits inside the triangle and is tangent to each side.

Circumcenter

The circumcenter is where all three perpendicular bisectors of the sides meet. Because it lies on every perpendicular bisector, it's equidistant from all three vertices. That equal distance becomes the radius of the triangle's circumscribed circle (circumcircle), the circle that passes through all three vertices.

Centroid

The centroid is where all three medians meet. A median connects a vertex to the midpoint of the opposite side. The centroid divides each median in a 2:1 ratio, with the longer segment always on the vertex side. In other words, the centroid sits $\frac{2}{3}$ of the way from any vertex to the midpoint of the opposite side.

Orthocenter

The orthocenter is where all three altitudes meet. An altitude runs from a vertex perpendicular to the opposite side (or the line containing the opposite side). Unlike the other centers, the orthocenter doesn't have a single clean equidistance property, but it plays a key role in advanced relationships like the Euler line.

Location of triangle centers

Where each center falls depends on the type of triangle. This is a common test topic, so know this table well:

For right triangles specifically, note the two special cases:

• The circumcenter lands exactly at the midpoint of the hypotenuse. This makes sense because the hypotenuse is the diameter of the circumscribed circle (a consequence of Thales' theorem).
• The orthocenter lands exactly at the vertex of the right angle, since the two legs themselves are altitudes.

Applications of triangle centers

Incenter and the inscribed circle

The incircle is tangent to each side at exactly one point. To find its radius, drop a perpendicular from the incenter to any side. All three perpendicular distances are equal, and that distance is the inradius.

There's also a handy area formula: for a triangle with area $A$, semiperimeter $s = \frac{a + b + c}{2}$, and inradius $r$, the relationship is $A = rs$. If you know the area and the side lengths, you can solve for the inradius directly.

Circumcenter and the circumscribed circle

The circumcircle passes through all three vertices. The distance from the circumcenter to any vertex is the circumradius. For a right triangle, the circumradius equals half the hypotenuse.

Centroid as the balance point

The centroid is the center of mass of a triangular region. If you cut a triangle out of uniform cardboard, it would balance perfectly on a pin placed at the centroid. The 2:1 median ratio is the reason: it distributes area evenly around that point.

Each median also divides the triangle into two smaller triangles of equal area. And all three medians together split the triangle into six smaller triangles that all have the same area.

Orthocenter and altitudes

The orthocenter connects to deeper geometry. The foot of each altitude is the point where the altitude meets the opposite side at a right angle.

The orthocenter, centroid, and circumcenter are always collinear, lying on the Euler line. The centroid divides the segment from the orthocenter to the circumcenter in a 2:1 ratio, with the centroid twice as close to the circumcenter as to the orthocenter. For an equilateral triangle, all four centers (incenter included) coincide at the same point, so the Euler line isn't defined separately.

Relationships between centers and segments

Incenter and angle bisectors

1. Construct the bisector of any two angles; their intersection is the incenter.
2. By the Angle Bisector Theorem, each bisector divides the opposite side into segments proportional to the two adjacent sides. For example, if the bisector of angle $A$ hits side $\overline{BC}$ at point $D$, then $\frac{BD}{DC} = \frac{AB}{AC}$.

Circumcenter and perpendicular bisectors

1. Construct the perpendicular bisector of any two sides; their intersection is the circumcenter.
2. Every point on a perpendicular bisector is equidistant from the two endpoints of that segment. The circumcenter, sitting on all three bisectors, is therefore equidistant from all three vertices.

Centroid and medians

1. Construct medians from any two vertices to the midpoints of the opposite sides; their intersection is the centroid.
2. The centroid divides each median in a 2:1 ratio (vertex to centroid : centroid to midpoint = 2 : 1). This ratio holds for every median in every triangle.

Orthocenter and altitudes

1. Construct the altitude from any two vertices perpendicular to the opposite side; their intersection is the orthocenter.
2. The foot of each altitude is the point of perpendicular intersection with the opposite side. In obtuse triangles, you may need to extend the opposite side to find where the altitude meets it, which is why the orthocenter ends up outside the triangle.