Ceva’s Theorem
Condition for three cevians to be concurrent.
Key idea
Condition for three cevians to be concurrent.
In depth
Lines from the vertices of a triangle to points D, E, F on opposite sides are concurrent exactly when (AF/FB)(BD/DC)(CE/EA) = 1. It provides a quick test for concurrency, including medians and altitudes.
Worked example
If AF/FB = 2 and BD/DC = 3, then CE/EA must be 1/6 for concurrency.
Formula
Real-world uses
- Checking that three cevians meet at one point
- Advanced problem-solving in geometry contests
Common mistake
Taking the ratios in inconsistent directions around the triangle.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Ceva’s Theorem?
Condition for three cevians to be concurrent. It is a topic at the expert level in Geometry.
What is the formula for Ceva’s Theorem?
The key formula is (AF/FB)(BD/DC)(CE/EA) = 1.
Where is Ceva’s Theorem used in real life?
Real-world examples: Checking that three cevians meet at one point; Advanced problem-solving in geometry contests.
What should I learn before Ceva’s Theorem?
Make sure you are comfortable with Nine-Point Circle, the previous topic in Geometry.
What is a common mistake with Ceva’s Theorem?
Taking the ratios in inconsistent directions around the triangle.
How can I practise Ceva’s Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Ceva’s Theorem?
Move on to Menelaus’ Theorem next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.