Areas in Triangle

In a triangle, mark 1/3 of each side counting from vertices in a certain order. Connect those points to the opposite vertices. Prove that the area of the middle triangle thus obtained is 1/7 that of the given triangle. This is a well-known puzzle with multiple solutions. One straightforward solution that makes use of Ceva's and VanObel's theorems generalizes to the case where we mark 1/N-th of each side. The illustration below draws on the knowledge derived from those two theorems.

(Note that for N = 2, we get what may be called a proof without words for the fact that the area of the triangle (shown in red) formed by the medians of a given triangle equals 3/4 of the area of the latter. Another such pww appeared in the April's issue (1999) of Mathematics Magazine. A trigonometric and similarity argument can be found in [Mathematics Visitor].

References

1. S. Rabinowitz (ed), Problems and Solutions from the Mathematical Visitor (1877-1896), MathPro Press, 1996, #8

Copyright © 1996-2009 Alexander Bogomolny