Wares’ (2007) three conjectures
Conjecture 1: In Figure 1 we have three mutually tangent semi-circles with centres
at collinear points X, Y, and Z. The radius of the semicircle with centre Z is a+b. The
rectangle ABCD is a by b. It seems as though the ratio of the area of the
arbelos(region bounded by three mutually tangent circles whose centres are
collinear) to the area of the rectangle is . Is this conjecture true?
Fig. 1
Conjecture 2: In figure 2, ABC is an arbitrary triangle. Points S, T, and U are the
feet of the medians from vertices A, B, and C to sides BC, AC, and AB, respectively.
Line segments QR, QP, and PR pass through points C, A, and B, respectively.
Moreover, line segments QR, QP and PR are drawn parallel to medians AS, BT, and
CU respectively. It seems as though the ratio of the area of ABC to the area of
PQR is 1/3. Is this conjecture true?
Figure 2
Conjecture 5: In figure 5, quadrilateral ABCD is a square and BCQ is quadrilateral.
The point X is the mid point of side CQ. We connect points A and X. It seems as
though the ratio of the length of the line segment BC to the length of the line
segment AX is the golden ratio . Is this conjecture true?
Figure 5
Varignon Theorem
The figure formed when the midpoints of the sides of a quadrilateral are joined in
order is a parallelogram. Equivalently, the bimedians bisect each other.
The area of the Varignon parallelogram of a quadrilateral is half that of the
quadrilateral, and the perimeter is equal to the sum of the diagonals of the original
quadrilateral.
The Gamow Problem
The pirate in our story has buried his treasure on the Greek island of Thasos and
noted its location on an old parchment: “You walk directly from the flag (point F) to
the palm tree (point P), counting your paces as you walk. Then turn a quarter of a
circle to the right and go to the same number of paces. When you reach the end, put
a stick in the ground (point K). Return to the flag and walk directly to the oak tree
(point O), again counting your paces and turning a quarter of a circle to the left and
going the same number of paces. Put another stick in the ground (point L). The
treasure is buried in the middle of the distance of the two sticks (point T).” After
some years the flag was destroyed and the treasure could not be found through the
location of the flag. Can you find the treasure now or is it impossible?”
The Original Gamow Problem
A young man was once thrilled to discover an old parchment bearing the following
clue to the whereabouts of a buried treasure.
"At XdegreesN and YdegreesW*, there lies a desert island where an old pirate once
buried his famous treasure, gathered during a lifetime of piracy. There on the north
shore, stands a solitary oak tree and nearby a solitary pine tree. Not far away, there
is also the gallows where we used to hang traitors.
Start from the gallows and walk towards the oak tree, counting your steps. At the
oak, turn right by a right angle and take the same number of steps. Here put in a
spike.
Now return to the gallows and walk towards the pine tree, counting your steps. At
the pine, turn left by a right angle and take the same number of steps. Here put in
another spike.
Dig half way between the spikes; the treasure is there.”
*Obviously I can't give you the real co-ordinates, otherwise you would dash off at
once to find the treasure.
Well, the young man (who had the correct map reference, of course), followed the
instructions and found the island and the two trees. Unfortunately, the gallows had
long-since disintegrated due to the weather, termites, etc, and no trace of its
location could be found. Sadly, the young man sailed away with nary a gold coin or
precious jewel for his pains. That is a great pity, because if you had been there, you
could have helped him find the treasure, couldn't you?
You could, couldn't you? How?
Activities used by Connor, Moss and Grover (2007)
1. If the incentre and the circumcentre of a triangle coincide, then the triangle is
an equilateral triangle.
2. If ABCD is a cyclical quadrilateral, then m(BAC) +m(BDC) = 180
3. If a cyclic quadrilateral is a parallelogram, then it is a rectangle
Activities used by Hadas, Hershkowitz, & Schwarz (2000)
Angles in Polygons
Task A: Measure (with the software) the sum of the interior angles in polygons as
the number of sides increases. Generalize, and explain your conclusion.
Task B: Measure (with the software) the sum of the exterior angles of a
quadrilateral. Hypothesize the sum of the exterior angles for polygons as the
number of sides increases. Check your hypothesis by measuring and explain what
you found.
Congruent Triangles
In this activity, we will investigate if and when, two triangles having several equal
elements, are congruent.
Part 1:
Task la: Given a dynamic ABC, construct another DEF having one angle and
one adjacent side equal to one angle and adjacent side of ABC. Drag the
vertices of both triangles and investigate the variance of the triangles and how
they relate one to another.
Task lb: Given a dynamic ABC, construct another DEF having two angles and
the included side equal to two angles and the included side of ABC. Drag the
vertices of both triangles and investigate the variance of the triangles and how
they relate one to another.
Task 2: Is it possible to construct a triangle with one side and two angles equal to
those of a dynamic ABC, but not congruent to ABC? If it is possible, construct
such a triangle; otherwise explain why.
Part 2:
Task 3a: How many equal sides and angles are there in the non-congruent
triangles constructed in task 2?
Task 3b: Is it possible to construct two non-congruent triangles, with five equal
elements (sides and angles)? Generate an hypothesis.
Task 4: Is it possible to construct two non-congruent triangles with six equal
elements? If yes, construct two such triangles, otherwise explain why.
Part 3:
Task 5a: Let's suppose that we already have 2 non-congruent triangles with five
equal elements, specify the equal elements.
Task 5b: Try to construct two non-congruent triangles, with five equal elements
Jones’ (2000, p. 65) phased activities
Phase 1
Produce identical figures to those given on paper , each involving lines and circles ,
which remain robust under drag.
Phase 2
Construct a rhombus, a square and a kite which are invariant under drag, explaining
why it works.
Phase 3
Use constructions to explain the relations between:
 The rhombus and the square
 The rectangle and the square
 The kite and the rhombus
 The parallelogram and the trapezium
 The rhombus, rectangle and parallelogram
Complete a hierarchical classification diagram for quadrilaterals using the phrase ‘is
a special case of’