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.

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.