**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.

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.

**Related Links**

Feynman's Triangle: Some Generalizations & Variations

Feynman Parallelogram Generalization

Area Parallelogram Partition Theorem

Some Parallelo-hexagon Area Ratios

Area ratios of some polygons inscribed in quadrilaterals and triangles

International Mathematical Talent Search (IMTS) Problem Generalized

A Geometric Paradox Explained (Another variation of an IMTS problem)

Euclid 1-43 Parallelogram Area Theorem

The theorem of Hippocrates (470 – c. 410 BC)

Triangle Area Formula in terms of angles, r & R

Cross's (Vecten's) theorem & generalizations to quadrilaterals

Area Formula for Quadrilateral in terms of its Diagonals

Maximum area of quadrilateral problem

The Equi-partitioning Point of a Quadrilateral

The Orthocentre Quadrilateral of a Quadrilateral

Minimum Area of Miquel Circle Centres Triangle

Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem

**External Links**

Richard Feynman (Biography at Wikipedia)

One-seventh area triangle

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Created by Michael de Villiers, Sept 2009; modified 26 July 2021; updated 9 Dec 2023.