Maxwell Theorem via the Center of Gravity

Michel Cabart
30 April, 2008

Below we offer a proof of Maxwell's theorem that is based on the notion of barycenter. Maxwell's theorem states the following fact:

Given ΔABC and a point G, the sides of ΔMNP are parallel to the cevians in ΔABC through G. Then the cevians in ΔMNP parallel to the sides of ΔABC are concurrent.

Below, the vector joining point A to B will be written in bold, so that, for example, AB = - BA.

In ΔABC with A', B', C' on sides opposite the vertices A, B, and C, the fact that the cevians AA', BB', CC' concur in point G is equivalent to either of the two conditions:

 There is a triple of real numbers (a, b, c), unique up to a non-zero factor, such that
(1)aGA + bGB + cGC = 0
 There is a triple of real numbers (a, b, c), unique up to a non-zero factor, such that
(2) A' = Z(B, b; C, c)
B' = Z(A, a; C, c)
C' = Z(A, a; B, b),

where Z(X, x; Y, y) denotes the barycenter of two material points X and Y with masses x at X and y at Y.

Let's suppose lines AA', BB' and CC' intersect.

Step 1: NP, PM, MN are parallel to GA, GB, GC means there exists x, y, z such that NP = xGA, PM = yGB, MN = zGC. As NP + PM + MN = 0, xGA + yGB + zGC = 0. A comparison with (1) shows that (x, y, z) is a multiple of (a, b, c). We can assume (x, y, z) = (a, b, c). Thus NP = aGA, PM = bGB, MN = cGC.

Step 2: MM', NN', PP' are parallel to BC, AC, AB meaning there is (m, n, p) such that MM' = mBC, NN' = nAC, PP' = pAB. The first equality yields

MM' = m(GC - GB) = (m/c)MN + (m/b)MP

so that

M' = Z(N, 1/c; P, 1/b) = Z(N, b; C, c)

by multiplying by bc. Similarly,

N' = Z(M, a; P, c) and
P' = Z(M, a; N, b).

This proves the theorem thanks to (2).

Barycenter and Barycentric Coordinates

  1. 3D Quadrilateral - a Coffin Problem
  2. Barycentric Coordinates
  3. Barycentric Coordinates: a Tool
  4. Barycentric Coordinates and Geometric Probability
  5. Ceva's Theorem
  6. Determinants, Area, and Barycentric Coordinates
  7. Maxwell Theorem via the Center of Gravity
  8. Bimedians in a Quadrilateral
  9. Simultaneous Generalization of the Theorems of Ceva and Menelaus
  10. Three glasses puzzle
  11. Van Obel Theorem and Barycentric Coordinates
  12. 1961 IMO, Problem 4. An exercise in barycentric coordinates
  13. Centroids in Polygon
  14. Center of Gravity and Motion of Material Points
  15. Isotomic Reciprocity
  16. An Affine Property of Barycenter
  17. Problem in Direct Similarity
  18. Circles in Barycentric Coordinates
  19. Barycenter of Cevian Triangle
  20. Concurrent Chords in a Circle, Equally Inclined

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