# Menelaus from 3D

Invocation of 3D imagery often serves a powerful tool in proving theorems of plane geometry. Farther down the page there are links to several demonstrations of this idea. On November 28, 2010, Hubert Shutrick had a vision that rendered a one-liner proof of Menelaus' theorem.

Let three points F, D, and E, lie respectively on the sides AB, BC, and AC of ΔABC or their extensions. Assume just one or all three of the points lie on side extensions. Then the points are collinear iff

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### Proof (Hubert Shutrick)

Put up a flagpole AA' orthogonal to the plane and let B' and C' be the points where the orthogonal lines through B and C meet the plane A'DEF. If the heights of the verticals are h_{a}, h_{b} and h_{c}, possibly negative, then similar vertical right triangles give

h_{a}/h_{b} = AF/BF, h_{b}/h_{c} = BD/CD and h_{c}/h_{a} = CE/AE

which, when multiplied together, produce exactly the required identity.

Conversely, assume Menelaus' identity holds. Erect AA' and consider the plane A'EF. Define B' and C' as before, and D' the intersection of B'C' with BC. Then

h_{a}/h_{b} = AF/BF, h_{b}/h_{c} = BD'/CD' and h_{c}/h_{a} = CE/AE.

Comparing the product of the three with Menelaus' identity shows that

### 2D Problems That Benefit from a 3D Outlook

- Four Travellers, Solution
- Desargues' Theorem
- Soddy Circles and Eppstein's Points
- Symmetries in a Triangle
- Three Circles and Common Chords
- Three Circles and Common Tangents
- Three Equal Circles
- Menelaus from 3D
- Stereographic Projection and Inversion
- Stereographic Projection and Radical Axes
- Sum of Squares in Equilateral Triangle

### Menelaus and Ceva

- The Menelaus Theorem
- Menelaus Theorem: proofs ugly and elegant - A. Einstein's view
- Ceva's Theorem
- Ceva in Circumscribed Quadrilateral
- Ceva's Theorem: A Matter of Appreciation
- Ceva and Menelaus Meet on the Roads
- Menelaus From Ceva
- Menelaus and Ceva Theorems
- Ceva and Menelaus Theorems for Angle Bisectors
- Ceva's Theorem: Proof Without Words
- Cevian Cradle
- Cevian Cradle II
- Cevian Nest
- Cevian Triangle
- An Application of Ceva's Theorem
- Trigonometric Form of Ceva's Theorem
- Two Proofs of Menelaus Theorem
- Simultaneous Generalization of the Theorems of Ceva and Menelaus
- Menelaus from 3D
- Terquem's Theorem
- Cross Points in a Polygon
- Two Cevians and Proportions in a Triangle, II
- Concurrence Not from School Geometry
- Two Triangles Inscribed in a Conic - with Elementary Solution
- From One Collinearity to Another
- Concurrence in Right Triangle
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Copyright © 1996-2017 Alexander Bogomolny

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