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?
A picture is made by joining five small quadrilaterals together to
make a large quadrilateral. Is it possible to draw a similar
picture if all the small quadrilaterals are cyclic?
This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.
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?
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?
A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4.
Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
A triangle PQR, right angled at P, slides on a horizontal floor
with Q and R in contact with perpendicular walls. What is the locus
ABCD is a rectangle and P, Q, R and S are moveable points on the
edges dividing the edges in certain ratios. Strangely PQRS is
always a cyclic quadrilateral and you can find the angles.
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.
Four rods are hinged at their ends to form a quadrilateral with
fixed side lengths. Show that the quadrilateral has a maximum area
when it is cyclic.
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.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
As a quadrilateral Q is deformed (keeping the edge lengths constnt)
the diagonals and the angle X between them change. Prove that the
area of Q is proportional to tanX.
Explore the shape of a square after it is transformed by the action
of a matrix.
Find the exact values of some trig. ratios from this rectangle in
which a cyclic quadrilateral cuts off four right angled triangles.