Menelaus's Theorem Calculator
Check if three points on triangle sides are collinear
Segment Lengths
On side AB:
On side BC:
On side CA:
Results
Individual Ratios
Step-by-Step Verification
Understanding Menelaus's Theorem
Menelaus's theorem (named after Menelaus of Alexandria, c. 100 AD) provides a necessary and sufficient condition for three points to be collinear when they lie on the sides (or extensions) of a triangle.
The Theorem
Points D, E, F lying on sides BC, CA, AB (or their extensions) are collinear if and only if the product of the three ratios equals 1.
Signed vs Unsigned Ratios
The theorem is typically stated using signed ratios (directed segments):
- Positive ratio: segments point in the same direction
- Negative ratio: segments point in opposite directions
This calculator uses unsigned (positive) ratios for simplicity. For the signed version, you may need to adjust signs based on point positions.
Key Points
- The theorem works for points on side extensions, not just on the sides themselves
- It's the converse of Ceva's theorem (concurrent cevians)
- Named after Greek mathematician Menelaus of Alexandria (c. 70-140 AD)
- Essential in projective geometry and proving collinearity
Frequently Asked Questions
What does collinear mean?
Collinear means that three or more points lie on the same straight line. Menelaus's theorem provides a way to verify this condition for points on triangle sides.
Who was Menelaus of Alexandria?
Menelaus was a Greek mathematician and astronomer who lived around 70-140 AD. He made important contributions to spherical geometry and trigonometry, particularly in his work "Sphaerica."
What's the difference between Menelaus and Ceva's theorems?
Menelaus's theorem checks if points are collinear (on a straight line), while Ceva's theorem checks if cevians are concurrent (meet at a point). They are dual theorems with similar structures.
Can points be on side extensions?
Yes! Menelaus's theorem works for points on the sides of the triangle or on their extensions. This is one reason the theorem uses signed (directed) ratios in its formal statement.
Why use signed ratios?
Signed ratios account for direction. If point F is between A and B, AF and FB point in the same direction (positive). If F is outside AB, they point in opposite directions (one is negative).
How accurate should the product be?
In theory, the product should exactly equal 1. In practice, due to rounding and measurement errors, we accept values very close to 1 (within a small tolerance, like 0.0001).
What are practical applications?
Menelaus's theorem is used in geometry proofs, computer graphics (determining if points align), surveying, and solving competition mathematics problems involving collinearity.
Is there a 3D version?
Yes! Menelaus's theorem has been generalized to three dimensions for tetrahedra and higher dimensions for simplices, checking when points lie on the same plane or hyperplane.