I am grateful to Harry Wiggens from the Dept. of Mathematics & Applied Mathematics, University of Pretoria, for bringing the following interesting problem to my attention. Harry is also an alumnus of the South African Mathematics Olympiad, finishing in the top ten in the country twice, as well as representing South Africa at the International Mathematical Olympiad (IMO) in 2001 and 2002.
Unfortunately the original source of the problem is not known. However, it provides a lovely example for use with dynamic geometry software.
Attached Regular Pentagons form Congruent Equilateral Triangles
Given any regular pentagon ABCDE with another regular pentagon IFGHD attached to the first pentagon at point D as shown. Next construct two equilateral triangles XAF and GBY as shown. Prove that these two equilateral triangles are congruent.
Investigate
Obviously to prove these two equilateral triangles are congruent, it suffices to show that a pair of corresponding sides are equal.
1) The dynamic sketch below illustrates that AF = BG. Investigate further by dragging.
Challenge
2) Can you explain why (prove that) the result is true?
(Hint: Click on the 'Hint' button on the bottom left and compare triangles DAF and DBG.)
3) Can you prove it in more than one way?
Attached Regular Pentagons form Congruent Equilateral Triangles
Explore More
Often understanding why a result is true, enables one to easily vary or further generalize the result.
4) Can you think of some variations or generalizations of the result above?
5) Check your answer to 4), by clicking on the 'Link to hexagon' and 'Link to octagon' buttons to check & compare some possible generalizations.
6) In the attached regular octagons sketch, also click the 'Show Lines' button. What do you notice?
7) Can you explain why (prove that) your observations in 5) and 6) are true?
8) Also explore the locus of S as point X is dragged (to vary the yellow octagon).
9) Can you explain (prove) your observation in 8)?
10) Click on the 'Link to semi-regular cyclic hexagon' button to navigate to a new sketch showing two similar semi-regular hexagons attached at a vertex. What do you notice?
11) Can you explain (prove) your observation in 10)?
12) Consider other variations and generalizations of the problem and explore further.
Published Paper
Read my paper entitled Attached Regular Pentagons produce Congruent Equilateral Triangles regarding this problem, and some variations, that has been published in the AMESA journal Learning and Teaching Mathematics, June 2025, no. 38, pp. 25-29.
Related Links
Clough's Theorem (a variation of Viviani) and some Generalizations
The Parallel-pentagon and the Golden Ratio
International Mathematical Talent Search (IMTS) Problem Generalized
A Geometric Paradox Explained
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Intersecting Circles Investigation
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
Anele Clive Moli's Method: Constructing an equilateral triangle
A 1999 British Mathematics Olympiad Problem and its dual
Dirk Laurie Tribute Problem
Golden Quadrilaterals
Extangential Quadrilateral
Triangulated Tangential Hexagon theorem
Theorem of Gusić & Mladinić
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Pirate Treasure Hunt and a Generalization
A Quarter Circle Investigation, Explanation & Generalization
Matric Exam Geometry Problem - 1949
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.)
***************
Free Download of Geometer's Sketchpad
***************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created with WebSketchpad by Michael de Villiers, 12 January 2025; modified 25 February 2025; updated 11 June 2025..