Draw a triangle, then draw three lines from its corners to the opposite sides. The odds of all three crossing at a single point might seem slim, but Ceva's Theorem explains exactly when they will.
Named after the Italian mathematician Giovanni Ceva, who published it in 1678, the theorem relies on lines called "cevians." A cevian is any line segment connecting a triangle's vertex to its opposite side. Ceva's Theorem states that three cevians will perfectly intersect at a single point—a geometric property called being "concurrent"—if and only if they divide the triangle's outer edges in a precise mathematical balance.
Specifically, if you move around the perimeter of the triangle and multiply the ratio of the two divided segments on the first side by the ratio of the segments on the second side, and multiply that by the ratio on the third side, the final result must equal exactly 1.
Ceva's Theorem acts as a unifying tool in Euclidean geometry. Instead of relying on complicated, unique proofs for different internal intersections, mathematicians use this single theorem to prove that:
- Medians (lines cutting the opposite side exactly in half) always meet at the triangle's centroid, or center of gravity.
- Altitudes (lines dropping at perfectly right angles to the opposite side) always meet at the orthocenter.
- Angle bisectors (lines slicing the corner angles exactly in half) always meet at the incenter.
While the mathematical world credits Giovanni Ceva, the proof was actually discovered in the 11th century by Yusuf al-Mu'taman ibn Hud, an Islamic king and geometer who ruled the Spanish city of Zaragoza. His mathematical text was lost for centuries, leaving Ceva to rediscover the principle independently 600 years later.