Unit 5: Relationships Within Triangles · ~15–30 min
Identify a triangle's four special centers and apply the centroid's 2:1 ratio to find median lengths.
Every triangle has four families of special segments, and each family always meets at a single point — a point of concurrency. The table below is the map: which segment, which point, which guarantee.
| Segment | Definition | Meeting point | Key property |
|---|---|---|---|
| Perpendicular bisectors | Bisects each side at 90° | Circumcenter | Equidistant from the three vertices; can fall inside, outside, or on the triangle |
| Angle bisectors | Bisects each angle | Incenter | Equidistant from the three sides; always inside the triangle |
| Medians | Vertex to the midpoint of the opposite side | Centroid | The triangle's balance point; always inside |
| Altitudes | Vertex, perpendicular to the opposite side (or its extension) | Orthocenter | Can fall inside, outside, or on the triangle |
The centroid deserves special attention because it comes with a numeric guarantee the others don't: it always divides each median into two pieces with a 2:1 ratio, measured from the vertex.
Centroid G divides median AM so that AG is twice GM.