Geometry • Topic 21

Ceva's Theorem

Ceva's Theorem is the primary tool for proving that three lines (cevians) inside a triangle are concurrent (meet at a single point).

Statement

Theorem (Ceva's Theorem)

Let

D, E, F

be points on sides

BC, C A, A B

△ A BC

. The lines

A D, BE, CF

are concurrent if and only if:

\frac{A F}{FB} \cdot \frac{B D}{D C} \cdot \frac{CE}{E A} = 1

Using the Law of Sines, Ceva's Theorem can be stated in terms of angles:

\frac{sin ∠ B A D}{sin ∠ D A C} \cdot \frac{sin ∠ A CF}{sin ∠ FCB} \cdot \frac{sin ∠ CBE}{sin ∠ EB A} = 1

"Go around the triangle." Start at a vertex, take the first segment, divide by the second. Multiply these ratios around the perimeter.

(A F / FB) \times (B D / D C) \times (CE / E A) = 1

Exercise (Problem 1)

Use Ceva's Theorem to prove that the three medians of a triangle are concurrent. (Hint: $D, E, F$ are midpoints, so ratios are all 1).

Exercise (Problem 2)

Use Ceva's Theorem to prove that the three internal angle bisectors are concurrent. (Hint: Use the Angle Bisector Theorem for the ratios).

Exercise (Problem 3)

△ A BC

A D, BE, CF

are concurrent at

P

. If

A F = 3, FB = 2, B D = 4, D C = 3

, find the ratio

A E / EC