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menelaus.md

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In a triangle ABC: D, E and F are three points on edges BC, CA and AB respectively. [1]

Menelaus's theorem states that D, E and F are collinear if and only if:

Ceva's theorem states that AD, BE and CF are concurrent if and only if:

When all points ABCDEF are written as homogeneous coordinates, we use some properties mentioned here (Trick 2c) and get:

Then we calculate each ratio:

Analogously, we have and . Then we get:

Take (A, B, C) as basis, then D, E and F are collinear if and only if:

i.e. n = -1, which proves Menelaus's theorem. [2]

To prove Ceva's theorem, we should calculate AD, BE and CF by Trick 3:

where (x, y, z) are coefficients on basis (A, B, C).

Then they are concurrent if and only if:

i.e. n = 1. [3]

Duality

It is not obvious that Menelaus's theorem and Ceva's theorem are dual. Let's start from this theorem:

Theorem 1 A straight line intersects a triangle ABC's three edges BC, CA and AB at point D0, E0 and F0 respectively. Take another three points D, E and F on BC, CA and AB respectively, then D, E and F are collinear if and only if:

A simple proof uses the lemma that if four different collinear points are represented in homogeneous coordinates as A, B, C=A+mB and D=A+nB (m and n are not zero), then . [4]

When we write D0, E0, F0, D, E and F as:

it's easy to prove that D0, E0 and F0 are collinear if and only if , and D, E and F are collinear if and only if . Then we get the product of three cross-ratios and finish the proof. [5]

When D0, E0 and F0 are at infinity, we get Menelaus's theorem.

The dual theorem states as:

Theorem 1' Three line d0, e0 and f0 passing through a triangle abc's 3 vertices A=b∩c, B=c∩a and C=a∩b respectively and meet at a point. Take another three lines d, e and f passing through A, B and C respectively, then d, e and f meet at another point if and only if:

When d0, e0 and f0 meet at the incenter, which means , and , we get another form of Ceva's theorem: [6]

Here is a similar explanation of the duality. Here is an explanation by barycentric coordinates.

Notes

  1. We use the diagrams from here and here.
  2. Here and here are proofs of Menelaus's theorem by Cartesian coordinates.
  3. Here and here are proofs of Ceva's theorem by Cartesian coordinates.
  4. Here is the proof.
  5. Here and here are vector space proofs.
  6. Here is the explanation.