Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for geometry triangle theorems centroid circumcenter? Then this Geometry lesson on Triangle Theorems and Triangle Centers explains each idea step by step, with worked examples.
Quick answer: Triangle interior angles sum to 180°, and an exterior angle equals the sum of its two remote interior angles. A midsegment is parallel to the third side and half as long. The medians meet at the centroid, two thirds of the way from each vertex. Perpendicular bisectors, angle bisectors and altitudes meet at the circumcenter, incenter and orthocenter.
Standards: G-CO.C.10, G-C.A.3 · Level: Grade 10 Geometry
What do the angles of a triangle add up to?
- Angle Sum: the three interior angles add to 180°.
- Exterior Angle Theorem: an exterior angle equals the sum of the two remote interior angles.
- Isosceles Triangle Theorem: angles opposite congruent sides are congruent, and the converse is true.
The exterior angle forms a linear pair with the interior angle at the same vertex, which is why it equals the sum of the other two interior angles.
What does a midsegment tell us?
A midsegment joins the midpoints of two sides of a triangle. It is parallel to the third side and half as long. The three midsegments cut a triangle into four congruent triangles.
Where do special segments of a triangle meet?
| Point | Made by | Key fact |
|---|---|---|
| Centroid | Medians (vertex to midpoint of opposite side) | Splits each median 2 : 1 from the vertex |
| Circumcenter | Perpendicular bisectors of the sides | Equidistant from the three vertices |
| Incenter | Angle bisectors | Equidistant from the three sides |
| Orthocenter | Altitudes (vertex perpendicular to opposite side) | Inside, on or outside the triangle |
What are inscribed and circumscribed circles?
The circumcenter is the center of the circumscribed circle, which passes through all three vertices. The incenter is the center of the inscribed circle, which touches all three sides. For a right triangle, the circumcenter is the midpoint of the hypotenuse.
- Centroid: average the three x-coordinates and the three y-coordinates.
- Circumcenter: find two perpendicular bisectors and intersect them.
- Check that the distances to the vertices are equal.
A median is 30 units long. The centroid is 2/3 of the way from the vertex, so it lies 2 · 30 ÷ 3 = 20 units from the vertex and 10 from the midpoint.
The 2 : 1 ratio belongs to the medians only. Angle bisectors and altitudes do not split each other in that ratio.
To remember the points: Centroid = Center of mass (medians), Circumcenter = Circle through vertices, Incenter = Inside circle.
- Interior angles of a triangle add to 180°; an exterior angle equals the sum of the remote interior angles.
- Base angles of an isosceles triangle are congruent, and congruent base angles mean congruent legs.
- A midsegment is parallel to the third side and half as long.
- The centroid divides each median 2 : 1; its coordinates are the averages of the vertices.
- The circumcenter is equidistant from vertices; the incenter is equidistant from sides.
Triangle Theorems and Triangle Centers: frequently asked questions
Why does the exterior angle equal the sum of the remote interior angles?
The exterior angle and its adjacent interior angle form a linear pair, so they add to 180°. The three interior angles also add to 180°, so the exterior angle must equal the other two interior angles together.
What is the difference between a median and an altitude?
A median runs from a vertex to the midpoint of the opposite side. An altitude runs from a vertex perpendicular to the line containing the opposite side. They coincide only in special triangles.
Where is the circumcenter of a right triangle?
It is the midpoint of the hypotenuse. The hypotenuse is a diameter of the circumscribed circle, so the midpoint is the same distance from all three vertices.
Can the orthocenter lie outside the triangle?
Yes. In an obtuse triangle the altitudes, extended as lines, meet outside the triangle. In a right triangle the orthocenter is the right-angle vertex.
How do you find the centroid from coordinates?
Average the x-coordinates of the vertices and average the y-coordinates. For vertices (0, 0), (6, 0) and (3, 9), the centroid is (3, 3).
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