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Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

### Long Short

A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot be greater than sqrt2. Find a quadrilateral of this type for which s2 is approximately sqrt3and show that s2 is always less than sqrt3. Find a quadrilateral of this type for which s3 is approximately 2 and show that s2 is always less than 2. Find a quadrilateral of this type for which s4=2 and show that s4 cannot be greater than 2.

### Diagonals for Area

Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.

##### Stage: 3 and 4

Cyclic quadrilaterals are quadrilaterals with all four of their vertices on a circle.

You can have cyclic polygons of any number of sides.

Not all quadrilaterals are cyclic. Perhaps you can draw a quadrilateral that is not cyclic - how do you know it is not cyclic?

All triangles are cyclic - how could you prove this?

Cyclic quadrilaterals have some interesting features and in this brief article we invite you to look at some of them and we suggest ideas for looking further:

Consider any cyclic quadrilateral $P Q R S$ with vertices on a circle centre $C$.

Draw chord $Q S$ and radii $P C$, $Q C$ and $S C$.

Let angle $Q P C=x$ degrees and angle $S P C=y$ degrees.

This means that angle $Q P S=(x+y)$ degrees.

The angles of the isosceles triangle $P Q C$ are $x$ degrees, $x$ degrees and $(180-2x)$ degrees.

The angles of the isosceles triangle $P S C$ are $y$ degrees, $y$ degrees and $(180-2y)$ degrees.

Angle $Q C S=360$ degrees $-$ angle $Q C P -$ angle $S C P=(2x+2y)$ degrees.

This shows that angle $Q C S$ is twice angle $Q P S$.

We say that angle $Q P S$ is subtended by the arc $Q R S$ and this basic property leads to the following theorems.

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.

Theorem 1.

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.

Theorem 2.

All angles in the same segment of a circle are equal (that is angles at the circumference subtended by the same arc).

Theorem 3.

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.

Theorem 4.

Opposite angles of a cyclic quadrilateral add up to 180 degrees. See this problem for a practical demonstration of this theorem.