Ceva's theorem is a theorem about triangles in Euclidean plane geometry. It regards the ratio of the side lengths of a triangle divided by cevians.
Menelaus's theorem uses a very similar structure. Both theorems are very useful in Olympiad geometry.
Ceva's theorem is useful in proving the concurrence of cevians in triangles and is widely used in Olympiad geometry.
Given a triangle with a point inside the triangle, continue lines to hit at respectively.
Ceva's theorem states that
The converse of Ceva's theorem is also true: If are on sides respectively such that , then lines are concurrent at a point .
There are various proofs for Ceva's theorem. In this wiki, we're going to prove it by using triangle's area.
because and have the same altitudes (and ditto for the last two triangles).
By subtracting the triangle areas of the second equality with the first equality, we get
Multiplying the previous three equations together, we get
Prove that if are midpoints of the sides, the three cevians are concurrent.
In this case, are the midpoints of their respective sides. Therefore, and so it immediately satisfies Ceva's theorem, demonstrating the existence of the intersection, which is the centroid.
Prove that cevians perpendicular to opposite sides are concurrent.
Let be the feet of the altitudes.
Note that and, similarly, . Therefore,
Multiplying these three equations gives us
Rearranging the left-hand side gives us
Therefore, the three altitudes coincide at a single point, the orthocenter.