Darts and Kites

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

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?

Rhombus in Rectangle

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Center Path

Stage: 3 and 4 Challenge Level:

The following solution comes from Daniel from Northumberland.

Point X moves around the circumference of a circle of diameter AB. This happens because the four angles at the centre of the rhombus (where the diagonals cross) are always 90 degrees each, no matter where X may be. The angle AXB is always a right angle except when X is at A or at B. The points C, X and A are on top of each other when the path of X goes through A, and similarly D, X and B coincide when X goes through B.