## Feynman Triangle & Parallelogram Variations

Triangle variation: For a triangle ABC in the plane, if each side is divided by a point (1/p), (measured say anti-clockwise), find a formula relating the area of ABC with that of A'B'C' (triangle formed by these points). Drag the slider p to investigate.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Feynman Triangle & Parallelogram Variations

Parallelogram variation: For a parallelogram ABCD in the plane, if each side is divided by a point (1/p), (measured say anti-clockwise), find a formula relating the area of ABCD with that of A'B'C'D' (parallelogram formed by these points). Click on the 'Link to ...' button in the above sketch and drag the slider p to investigate.

Challenge
What formulae did you find? Can you prove your findings? Can you prove them in more than one way?

Further Generalization
Can you generalize further if the sides of the triangle & parallelogram are divided into different ratios?

Reference
De Villiers, M. (2005). Feedback: Feynman's Triangle (extended). The Mathematical Gazette, 89 (514), March, p. 107.

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Created by Michael de Villiers, Sept 2009; modified 26 July 2021.