## Parallelogram Distances

The following little investigation is suitable for learners at approximately Grades 8-9. It is a 'Further Explore' activity from my "Rethinking Proof with Sketchpad" book, which is available at McGraw-Hill in the USA, or internationally at Amazon, or other resellers.
Explore
1) ABCD is a parallelogram. Drag point P and observe the sum of the (perpendicular) distances from P to the sides. What do you notice?
2) Drag any of the red vertices of ABCD and then repeat step 1).
3) Drag P outside ABCD. What do you notice about the sum of the distances?
4) Formulate a conjecture on the basis of your observations.

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Explanation
5) Can you logically explain why (prove that) your observations in 4) above is true? (Note: You may not assume that the paths FPG and HPI are straight (even though they may appear to be so, you have to prove they are straight).
6) Google the concept of 'directed distances' and use it to ensure that your conjecture and proof also holds when P is outside the parallelogram.
7) Can you further generalize to hexagons, octogons, etc.?
8) In regard to 7), see the following related results, 2D Generalizations of Viviani's Theorem.

Read my paper The Value of using Signed Quantities in Geometry in Learning & Teaching Mathematics, Dec 2020.

Converse
9) Further investigate the converse of the parallelogram result, namely: If the sum of the (perpendicular) distances from an arbitrary point P to the sides of a quadrilateral is constant, then the quadrilateral is a parallelogram.
10) If true, can you prove it? If false, can you provide a counter-example?
11) Check your answer to 9) & 10) here: Viviani's Theorem: Parallelogram Extension.

Created by Michael de Villiers, 23 July 2020; updated 8 March 2021.