Lesson 5 of 14, 12 min read
Bisectors, medians, altitudes and triangle centers
Perpendicular bisectors, angle bisectors, medians and altitudes of a triangle, and the four points of concurrency: circumcenter, incenter, centroid and orthocenter.
Four kinds of special segments
Each triangle has three of each. Remarkably, for each kind, the three always meet at a single point. When three or more lines meet at one point, they are concurrent, and the point is a point of concurrency.
Two useful theorems
These explain the special properties of two of the triangle centers.
The four centers
| Lines | Point of concurrency | Key property |
|---|---|---|
| perpendicular bisectors | circumcenter | equidistant from the three vertices; center of the circumscribed circle |
| angle bisectors | incenter | equidistant from the three sides; center of the inscribed circle |
| medians | centroid | located \(\frac{2}{3}\) of the way from each vertex to the opposite midpoint |
| altitudes | orthocenter |
The centroid
The centroid is also the triangle's balance point: a cardboard triangle balances on a pencil tip placed there.
Where the centers lie
The incenter and centroid are always inside the triangle. The circumcenter and orthocenter move around:
| Triangle | Circumcenter | Orthocenter |
|---|---|---|
| acute | inside | inside |
| right | midpoint of the hypotenuse | vertex of the right angle |
| obtuse | outside | outside |
Practice
Quick check
The medians of a triangle meet at the:
The centroid is the point of concurrency of the medians.
Quick check
Which point is equidistant from the three vertices of a triangle?
The circumcenter lies on all three perpendicular bisectors, so it's equidistant from all three vertices.
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