Menelaus’ Theorem
Condition for three points on sides to be collinear.
Key idea
Condition for three points on sides to be collinear.
In depth
A line meeting the sides of a triangle (extended) at D, E, F satisfies (AF/FB)(BD/DC)(CE/EA) = −1 in signed lengths, or 1 in unsigned lengths. It tests collinearity, the partner of Ceva’s theorem.
Worked example
Three points on extended sides are collinear if the unsigned ratio product equals 1.
Real-world uses
- Checking that three points on triangle sides are collinear
- Used in map and survey problems
Common mistake
Mixing it up with Ceva; Menelaus is for collinear points.
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 Menelaus’ Theorem?
Condition for three points on sides to be collinear. It is a topic at the expert level in Geometry.
Where is Menelaus’ Theorem used in real life?
Real-world examples: Checking that three points on triangle sides are collinear; Used in map and survey problems.
What should I learn before Menelaus’ Theorem?
Make sure you are comfortable with Ceva’s Theorem, the previous topic in Geometry.
What is a common mistake with Menelaus’ Theorem?
Mixing it up with Ceva; Menelaus is for collinear points.
How can I practise Menelaus’ 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 Menelaus’ Theorem?
Move on to Ptolemy’s 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.