Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
Triangle ABC has equilateral triangles drawn on its edges. Points
P, Q and R are the centres of the equilateral triangles. What can
you prove about the triangle PQR?
Take any parallelogram and draw squares on the sides of the
parallelogram. What can you prove about the quadrilateral formed by
joining the centres of these squares?
Find the vertices of a pentagon given the midpoints of its sides.
A design is repeated endlessly along a line - rather like a stream
of paper coming off a roll. Make a strip that matches itself after
rotation, or after reflection
Two intersecting circles have a common chord AB. The point C moves
on the circumference of the circle C1. The straight lines CA and CB
meet the circle C2 at E and F respectively. As the point C. . . .
Two circles intersect at A and B. Points C and D move round one
circle. CA and DB cut the other circle at E and F. What do you
notice about the line segments CD and EF?
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Move the ends of the lines at points B and D around the circle and
find the relationship between the length of the line segments PA,
PB, PC, and PD. The length of each of the line segments is. . . .
The triangle OPA has a vertex O at the origin and OA along the x
axis, such that P has coordinates (x, y) and A has coordinates (2x,
0). By moving the position of the point P infinitely many. . . .
In the middle ages stone masons used a ruler and compasses method
to construct exact octagons in a given square window. Open your
compasses to a radius of half the diagonal of the square. . . .
Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
X is a moveable point on the hypotenuse, and P and Q are the feet
of the perpendiculars from X to the sides of a right angled
triangle. What position of X makes the length of PQ a minimum?
The points P, Q, R and S are the midpoints of the edges of a convex
quadrilateral. What do you notice about the quadrilateral PQRS as
the convex quadrilateral changes?
Points D, E and F are on the the sides of triangle ABC.
Circumcircles are drawn to the triangles ADE, BEF and CFD
respectively. What do you notice about these three circumcircles?
Four rods of equal length are hinged at their endpoints to form a
rhombus. The diagonals meet at X. One edge is fixed, the opposite
edge is allowed to move in the plane. Describe the locus of. . . .
The points P, Q, R and S are the midpoints of the edges of a
non-convex quadrilateral.What do you notice about the quadrilateral
PQRS and its area?
In the diagram the radius length is 10 units, OP is 8 units and OQ
is 6 units. If the distance PQ is 5 units what is the distance P'Q'
A circle has centre O and angle POR = angle QOR. Construct tangents
at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q
lie inside, or on, or outside this circle?
When a strip has vertical symmetry there always seems to be a
second place where a mirror line could go. Perhaps you can find a
design that has only one mirror line across it. Or, if you thought
that. . . .
Three equilateral triangles ABC, AYX and XZB are drawn with the
point X a moveable point on AB. The points P, Q and R are the
centres of the three triangles. What can you say about triangle
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Two perpendicular lines lie across each other and the end points
are joined to form a quadrilateral. Eight ratios are defined, three
are given but five need to be found.
In the diagram the point P can move to different places around the
dotted circle. Each position P takes will fix a corresponding
position for P'. As P moves around on that circle what will P' do?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?
In the diagram the point P' can move to different places along the
dotted line. Each position P' takes will fix a corresponding
position for P. If P' moves along a straight line what does P do ?
We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.
If I print this page which shape will require the more yellow ink?
I noticed this about streamers that have rotation symmetry : if
there was one centre of rotation there always seems to be a second
centre that also worked. Can you find a design that has only. . . .
Start with any triangle T1 and its inscribed circle. Draw the
triangle T2 which has its vertices at the points of contact between
the triangle T1 and its incircle. Now keep repeating this. . . .
The circumcentres of four triangles are joined to form a
quadrilateral. What do you notice about this quadrilateral as the
dynamic image changes? Can you prove your conjecture?
Two tangents are drawn to the other circle from the centres of a
pair of circles. What can you say about the chords cut off by these
tangents. Be patient - this problem may be slow to load.
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.