Roaming Rhombus

We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point D?

Center Path

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 point X and prove your assertion.

Rolling Coins

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a line of 'n' coins

Just Rolling Round

Stage: 4 Challenge Level:

 Tomas of Malmsbury School, Mark of The British School of Manilla and Herbert of Sha Tin College, Hong Kong sent good solutions to this problem. Well done all of you. $P$ is a point on the circumference of a circle radius $r$ which touches another circle radius $2r$ on the inside. The smaller circle rolls, without slipping, around the inner circumference of the larger circle. The point $P$ is a fixed point on the smaller circle which moves as the small circle moves. The point $P_o$ is the position of $P$ when $P$ is at the point of contact between the two circles. Consider the general position where the point of contact is the point $C$ but here we do not assume that $P_1$ is the position of the point $P$.
By showing that the lengths of the arcs $P_oC$ and $P_1C$ are equal, we shall prove that $P_1$ must be the position of the point $P$ when the point of contact is at $C$. Hence we shall show that $P$ must always lie on the diameter of the large circle through $OP_o$.

Let $M$ be the centre of the small circle, then $MO = MP_1 = MC = r$ and the triangle $OMP_1$ is isosceles. Hence $$\angle MOP_1 = \angle MP_1O = \theta$$ $$\angle P_1MC = \pi - (\pi - 2\theta) = 2\theta.$$ Hence, using the formula "arc length = radius x angle at the centre of the circle": $$P_0C = (2r)(\theta) = 2r\theta$$ and $$P_1C = (r)(2\theta) = 2r\theta.$$ Hence $P$ must be at the point $P_1$ because the circle rolls without slipping, which shows that the locus of P is the diameter of the larger circle.