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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.

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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?

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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: Challenge Level:1

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.