# Trigonometry/Circles and Triangles/Ceva's Theorem

**Ceva's Theorem** is as follows:

Let ABC be the vertices of a triangle. Let D be a point on side BC, E be a point on side AC and F be a point on side AB. (The points DEF may be on the extensions of the sides rather than the sides themselves.) Then the lines AD, BE, CF are **concurrent** (i.e. all cross at the same point) if and only if

It was discovered by Giovanni Ceva (1648-1734). Because of this theorem, any line joining the vertex of a triangle to a point on an opposite side is sometimes called a **cevian**.

## Some corollaries of Ceva's Theorem[edit]

1. The medians of a triangle are concurrent. (This is the centroid.)

2. The angle bisectors of a triangle are concurrent. (This is the incenter; this result has already been proved.)

3. The altitudes of a triangle are concurrent. (This is the orthocenter.)

4. (Gergonne's theorem) Let D, E, F be the points where the incircle touches the sides of the triangle ABC. Then the lines AD, BE and CF are concurrent. This theorem is due to Joseph Diaz Gergonne (1771-1859).

5. (Nagel's theorem) Let D, E, F be the points where the the respective excircles touch the sides AB, BC and CA respectively of the triangle ABC. Then the lines AD, BE and CF are concurrent.