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 now available for free as PDF to download at ResearchGate.
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.
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.
Published Paper
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.
Related Links
2D Generalizations of Viviani's Theorem
Further generalizations of Viviani's Theorem (using equi-inclined lines)
Clough's Theorem (a variation of Viviani)
Area Parallelogram Partition Theorem
Another parallelogram area ratio
Golden Parallelogram
Similar Parallelograms: A Generalization of a Golden Rectangle property
A generalization of a Parallelogram Theorem to Parallelo-hexagons, Hexagons and 2n-gons in general
Euclid 1-43 Parallelogram Area Theorem
Feynman Parallelogram Generalization
International Mathematical Talent Search (IMTS) Problem Generalized
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
Trio of Parallelograms
An associated result of the Van Aubel configuration and some generalizations
A Rectangle Angle Trisection Result
Crossed Quadrilateral Properties
A generalization of Neuberg's Theorem and the Simson line
Equi-inclined Lines Problem
A variation of Miquel's theorem and its generalization
Cyclic Hexagon Alternate Angles Sum Theorem
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
External Links
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Free Download of Geometer's Sketchpad
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Created by Michael de Villiers, 23 July 2020; updated 8 March 2021; 20 Oct 2024.