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

$\displaystyle {AF \over FB}\times {BD \over DC}\times {CE \over EA}=1.$

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

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 respective excircles touch the sides AB, BC and CA respectively of the triangle ABC. Then the lines AD, BE and CF are concurrent.