GeoProof — Ceva's Theorem explorer

(BD / DC) · (CE / EA) · (AF / FB) = 1

Drag P, or any vertex. The three cevians always pass through P, so the product of the ratios stays at exactly 1.

Ceva's Theorem

A cevian is a segment from a vertex of a triangle to a point on the opposite side. Take three cevians AD, BE, CF (with D on BC, E on CA, F on AB). Ceva's theorem says they all pass through one common point if and only if:

(BD / DC) · (CE / EA) · (AF / FB) = 1

In this tool the three cevians are built through a shared point P, so they are concurrent by construction — and the product stays pinned at 1 as you drag.

Worked example: the medians

If D, E, F are the midpoints of the sides, every ratio is 1, so the product is 1·1·1 = 1 — and indeed the three medians meet at the centroid. Drag P toward the centroid to see all three ratios approach 1.

Other concurrent sets

The same equation is satisfied by the three angle bisectors (meeting at the incenter) and the three altitudes (meeting at the orthocenter) — each is a special choice of D, E, F obeying Ceva's condition.

Signed version

Using signed ratios, the product equals +1, which also handles cevians meeting outside the triangle. With unsigned lengths and an interior point P, as shown here, the product is simply 1.

A companion: Menelaus

Ceva's theorem (about concurrence) has a twin, Menelaus' theorem (about three points being collinear), where the analogous signed product equals −1 instead of +1.

A bit of history

It is named for the Italian mathematician Giovanni Ceva, who published it in 1678 — though it was found earlier by the 11th-century Spanish-Arab mathematician Yusuf al-Mu'taman ibn Hud.