Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
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What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
Published March 1997,February 2011.
You might like to use the Geoboard environment and some of the problems that were published in July 2005 to help investigate these ideas practically before moving into the theory.
The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.
Because angle $Q C S$ is the same for all positions of $P$, Theorem 1 shows angle $Q P S$ is the same regardless of where $P$ lies.
See this problem for a practical demonstration of this theorem.
All angles in the same segment of a circle are equal (that is angles at the circumference subtended by the same arc).
The angle subtended by a semicircle (that is the angle standing on a diameter) is a right angle. See this problem for a practical demonstration of this theorem.
Opposite angles of a cyclic quadrilateral add up to 180 degrees. See this problem for a practical demonstration of this theorem.