Exploration
1) Start with any square ABCD and an arbitrary point E on AD (extended). Bisect ∠EBC with ray BF, and F on CD (extended).
2) Drag point E, and observe the measurements AE (a), CF and BE. Click on the 'Show Measurement' button. What do you notice?
Conjecture Formulate a conjecture, and check by dragging E along AD (also on to its extensions).
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
" ... even if we succeeded in finding a satisfactory solution we may still be interested in finding another solution. We desire to convince ourselves of the validity of a theoretical result by two different derivations as we desire to perceive a material object through two different senses. Having found a proof, we wish to find another proof as we wish to touach an object after having seen it. Two proofs are better than one. 'It is safe riding at two anchors'". - George Polya (1945: 61-62), How to Solve It.
Explanation (proof)
Can you explain why (prove) your conjecture above is true? If so, can you find other, different ways of logically explaining (proving) your conjecture?
Remark
A special case of this theorem was used for the Senior SA Mathematics Olympiad 2016, Round 1, Question 20, where AE = 2 and CF = 3 was given and learners were asked to determine BE. The 2016 Question Paper, Round 1 with Solutions are available at: SAMO 2016 R1 Paper & Solutions. (Papers and solutions for other SAMO rounds, and years, can be found at the relevant link in the External Links below).
Related Links
SA Mathematics Olympiad 2016 Problem R2 Q20
SA Mathematics Olympiad 2022, Round 2, Q25
IMO 2014 Problem 4 - Geometry
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Some Quadrilateral Inequalities involving Sides & Diagonals (1st Th. was used as Q1 in SAMO 2001, R3)
Matric Exam Geometry Problem - 1949 (A variation of Reim's theorem)
Dirk Laurie Tribute Problem
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
Vreken's sum of triangular areas in a regular hexagon (Application of Viviani)
Varignon Parallelogram Area (Rethinking Proof activity)
Parallel Lines (Thomsen's Hexagon) (Rethinking Proof activity; incl. of generalization)
Area Parallelogram Partition Theorem
International Mathematical Talent Search (IMTS) Area Ratio Problem Generalized
Another parallelogram area ratio
Feynman's Triangle: Some Generalizations & Variations
Cross's theorem & generalizations to quadrilaterals
Hippocrates' Theorem (470 – c. 410 BC)
Some Parallelo-hexagon Area Ratios
Easy Hexagon Explorations
Area Formula for Quadrilateral in terms of its Diagonals
The Area of an Orthodiagonal quadrilateral
Bicentric Polygon Area Formula in terms of angles, r & R
Cyclic Quadrilateral Difference of Squares Theorem
A Geometric Paradox Explained (Another variation of an IMTS problem)
External Links
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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.)
********************************
Back to Dynamic Geometry Sketches
Created by Michael de Villiers with WebSketchpad, 26 April 2016; updated 7 Sept2026.