P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
To find the point, try the following experiment:
The Thought Experiment
Consider any triangle where the angles are all less than 120
degrees. Imagine 3 pieces of string of equal length and knot them
together so that the 3 trailing ends are of equal length. Imagine a
tiny hole at each vertex of the triangle and thread the ends of the
string through the 3 holes. Attach 3 equal weights to the ends and
support the triangle horizontally so that the weights hang down
freely. What happens to the knot?
Continuing the 'thought experiment': The knot slides across the
triangle until the system settles down in equilibrium with the
tension in all three strings equal. Because of these equal tensions
the angles between the 3 strings must all be 120 degrees. Because
the weights will drop as low as they can the sum of the lengths of
the hanging pieces of the strings will be a maximum and the sum of
the lengths of the pieces from the knot to the vertices will be a
minimum (minimising the potential energy of the system).
For triangles where the biggest angle is 120 degrees or more the
knot will settle at the vertex with the obtuse angle.
To construct the point draw an equilateral triangle on each side
of the original triangle ABC and then join the outer vertices of
the equilateral triangles to the opposite vertices of the original
triangle and you will find that these three lines are concurrent
(i.e. they all go through one point). Label this point O. You will
find that the angles between OA and OB, between OB and OC and
between OC and OA are all 120 degrees. The proof of these facts
Constructing the point