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Maxwell's theorem (geometry)

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Given a triangle and a point, constructs a second triangle with a special point
Line segments with identical markings are parallel.
If the sides of the triangle A ′ B ′ C ′ {\displaystyle A'B'C'} are parallel to the according cevians of triangle A B C {\displaystyle ABC} , which are intersecting in a common point V ′ {\displaystyle V'} , then the cevians of triangle A ′ B ′ C ′ {\displaystyle A'B'C'} , which are parallel to the according sides of triangle A B C {\displaystyle ABC} intersect in a common point V ′ {\displaystyle V'} as well

Maxwell's theorem is the following statement about triangles in the plane.

For a given triangle A B C {\displaystyle ABC} and a point V {\displaystyle V} not on the sides of that triangle construct a second triangle A ′ B ′ C ′ {\displaystyle A'B'C'} , such that the side A ′ B ′ {\displaystyle A'B'} is parallel to the line segment C V {\displaystyle CV} , the side A ′ C ′ {\displaystyle A'C'} is parallel to the line segment B V {\displaystyle BV} and the side B ′ C ′ {\displaystyle B'C'} is parallel to the line segment A V {\displaystyle AV} . Then the parallel to A B {\displaystyle AB} through C ′ {\displaystyle C'} , the parallel to B C {\displaystyle BC} through A ′ {\displaystyle A'} and the parallel to A C {\displaystyle AC} through B ′ {\displaystyle B'} intersect in a common point V ′ {\displaystyle V'} .

The theorem is named after the physicist James Clerk Maxwell (1831–1879), who proved it in his work on reciprocal figures, which are of importance in statics.

References

  • Daniel Pedoe: Geometry: A Comprehensive Course. Dover, 1970, pp. 35–36, 114–115
  • Daniel Pedoe: "On (what should be) a Well-Known Theorem in Geometry." The American Mathematical Monthly, Vol. 74, No. 7 (August – September, 1967), pp. 839–841 (JSTOR)
  • Dao Thanh Oai, Cao Mai Doai, Quang Trung, Kien Xuong, Thai Binh: "Generalizations of some famous classical Euclidean geometry theorems." International Journal of Computer Discovered Mathematics, Vol. 1, No. 3, pp. 13–20

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