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This essential trains for: Math Kangaroo 7-8, Math Kangaroo 9-10, AMC-10, AMC-12, AIME.

Cyclic quadrilaterals can be inscribed in a circle. The vertices of the quadrilateral are points on the circle.

figure

Theorem: The opposite interior angles are supplementary (i.e. their sum, in degrees, is equal to 180°.) In the figure, the pairs of supplementary angles have been marked with the same color.

Proof: The angles of the quadrilateral are inscribed in the circle. The measure of an inscribed angle is half the measure of the intercepted arc:

equation

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Identify pairs of angles that are necessarily congruent, as they intercept the same arc:

figure


In cyclic quadrilaterals, Ptolemy's theorem applies:

equation


Fact: With a given set of four side lengths, it is always possible to form a cyclic quadrilateral.

Place three of the segments on a line:

figure

Now 'bend' the line into a circle, and reduce the radius (increase the curvature) until the 4-th side becomes a chord.

Of all the quadrilaterals with the same set of side lengths, the cyclic one has the largest area.


Brahmagupta's theorem is used to compute the area of a cyclic quadrilateral given the lengths of its sides: a,b,c,d.

First, define the semi-perimeter:

equation

The area is given by:

equation