F.7 Ceva and Menelaus

Concurrent Cevians, Collinear Transversals, the MNIJ Lattice Framework, and Cevian Programs

Three cevians AD, BE, CF of triangle ABC concur at a point P if and only if (AF/FB)(BD/DC)(CE/EA) = +1 (Ceva’s Theorem), whereas points D, E, F’ on the sidelines are collinear if and only if (AF’/F’B)(BD/DC)(CE/EA) = -1 (Menelaus’s Theorem), making (A, B; F, F’) a harmonic range.
ABC DEF c a b AD BE CF Menelaus AB_ext A (0.000, 0.000) B (7.000, 0.000) C (4.286, 4.199) P (4.238, 1.866) D (5.449, 2.400) E (2.857, 2.799) F (4.200, 0.000) F' (21.000, 0.000) AA BB CC PP DD EE FF F'F' + − ⟲ Lbl Gde

Interactive GEOMETOR construction for F.7 (Ceva and Menelaus). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point P : Point ===== D = intersect(line(A, P), line(B, C)) E = intersect(line(B, P), line(C, A)) F = intersect(line(C, P), line(A, B)) F’ = intersect(line(D, E), line(A, B))

Mathematical Synthesis

Spanning 26 folios (SeeligNotebooks), Topic 07 is one of the richest investigations in the Seelig archive. Combining Ceva's theorem (Giovanni Ceva, \(+1\)) and Menelaus's theorem (Menelaus, \(-1\)), Seelig demonstrates that the cevian trace F and the trilinear polar intersection F divide side A B in a harmonic conjugate range \(AF/FB = AF'/BF'\). In Folio 1 (FWS 2011/11/07), he applies Ceva's Theorem to equal tangent segments \(x = s-a, y = s-b, z = s-c\) to prove the concurrency of the Gergonne point \(G_e = X(7)\).

Across Folios 2–22, Seelig develops the MNIJ coordinate system (\(A=(0,0), B=(M,0), C=(I,J)\), with \(y\)-intercept \(N = JM/(M-I)\)), deriving closed-form rational expressions for the Cartesian trace coordinates D, E, F, the cevian lengths \(d, e, f\) via Stewart's theorem, the trilinear area of the cevian triangle \(\triangle DEF\), and a complete HP-35s calculator program (FWS 8/25/2014) verified on \((a,b,c) = (6,9,13)\) and on a Heronian \((78, 90, 84)\) triangle (April 14, 2015).

Key Algebraic Identities

\[\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = +1 \quad (\text{Ceva}), \qquad \frac{AF'}{F'B} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = -1 \quad (\text{Menelaus})\]
\[D_x = \frac{M J P}{J P + (M - I) Q}, \qquad E_x = \frac{M I Q}{I Q + (M - P) J}, \qquad F_x = \frac{I Q - J P}{Q - J}\]
\[\frac{PD}{AD} + \frac{PE}{BE} + \frac{PF}{CF} = 1, \qquad \frac{AP}{AD} + \frac{BP}{BE} + \frac{CP}{CF} = 2\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios

F.7 Ceva and Menelaus — Folios (Folios 0024–0049, 26 scanned manuscript pages in resources/seelig/07_ceva-and-menelaus)

Euclid’s Elements

VI.2 (EuclidElements)

GEOMETOR Elements (G-Index)

G.3 extremities of a segment, G.5 surface

Encyclopedia of Triangle Centers

X(1) — Incenter, X(2) — Centroid, X(7) — Gergonne Point, X(8) — Nagel Point (Kimberling1998)

Seelig Spreadsheets

Ceva & Menelaus, Ceva, Menelaus, Ceva Compare, 6-9-13 Cevian (Spreadsheets)

Computational Workbook Verification

The following worksheets from seelig’s Triangles and Circles.ods workbook verify the numerical and symbolic formulas for this topic on the canonical \((a, b, c) = (5, 6, 7)\) triangle:

Worksheet: Ceva (Triangles and Circles.ods)

Key Formulas & Parameters

Cell

Label / Context

Value

OpenFormula

F33

A

-8

=-F25

I33

I33

-8

=F38

J33

-8

0

=G38

F34

B

16

=F27

I34

I34

16

=F39

J34

16

0

=G39

G35

0

8

=F26

I35

I35

0

=F40

Cell Excerpt

Ceva’s Theorem

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If three lines meet inside a triangl

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Triangle ABC. Create three points D,

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for all possible triangles, then K i

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The product of side ratios equals un

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(AF/BF)*(BD/CD)*(CE/AE) = 1

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Worksheet: Menelaus (Triangles and Circles.ods)

Key Formulas & Parameters

Cell

Label / Context

Value

OpenFormula

J10

J10

5

=G10

K10

5

6

=H10

J11

J11

0

=G11

K11

0

0

=H11

J12

J12

18

=G12

K12

18

0

=H12

D13

D13

A

=D10

G13

G13

5

=G10

Cell Excerpt

Menelaus Theorem

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A line cuts the triangle ABC at thre

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The product of ratios (BD/DC)*(CE/EA

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Fred Seelig

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Begun Oct. 28, 2011

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July 5, 2012

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Constructive Dependency Graph

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Prerequisites: F.2 Bilinear Transformations, F.6 Centroid

Required for: F.23 Golden Triangles, F.39 Pole and Polar with Respect to a Triangle, F.46 Stewart’s Theorem, F.47 The UVW Coordinate Triangle