MATHSEARCH 2000
Senior Project Prepared by Geoff Ball and Humphrey Gastineau-Hills, University of Sydney.

Pedan Polygons and Polyhedra - page 2

Definition: An isosceles trapezium is one whose parallel sides are unequal and whose non parallel sides are equal.

Exercise 1 (6 marks):
For an isosceles trapezium ABCD with AB DC and AB<CD, prove that:
(i) ADC = BCD. Hint: Draw AX parallel to BC to meet DC in X.
(ii) ABCD is a cyclic quadrilateral;
(iii) the diagonals of ABCD are equal.

Exercise 2 (6 marks):
Suppose an isosceles trapezium has:
(a) parallel sides of lengths a and b with a < b;
(b) non-parallel sides of lengths c and
(c) the diagonals of length d
Use Ptolemy's theorem to prove that d² - c² = ab. We will call this Pedan's formula.
Notice that there are restrictions on the terms in Pedan's formula.
Importantly, the lengths a, b, c and d are constrained by the triangle inequality (that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. )

Exercise 3 (8 marks):
(i) Find the Pedan isosceles trapezium ABCD with BC=AD= 2 and for which p⁺ < 35.

(ii) Find the Pedan isosceles trapezium ABCD with BC=AD=4 and for which p⁺ < 35.

Generally, finding Pedan pentagons, hexagons and higher order polygons requires considerable insight and patience though one can get the flavour of such investigations using congruent triangles to form isosceles trapezia, for instance, which in turn can be organised into Pedan hexagons.

The following hexagon, known to famous mathematicians such as Leonard Euler, is a Pedan hexagon.

Suppose we begin with the triangle P₁P₂P₃ in which:

(a) P₁P₂ P₃ = 120°, and
(b) P₁P₂ = 3 and P₂P₃ = 5, respectively

Now, on the same side of P₂P₃ as P₁ is, construct P₂P₃P₄ P₃P₂P₁.

Note the order of the corresponding vertices which ensures this triangle is unique.

Complete the convex hexagon P₁P₂P₃P₄P₅P₆ using the same congruent triangles with angles of 120° at P₄ and P₅ respectively.

We hypothesise that the hexagon so formed is a Pedan hexagon!

Exercise 4. (2 marks) Use the cosine rule to show that the congruent triangles are Pedan.

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