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Diagonals of an inscribed quadrilateral.
Ptolemy's Theorem
Given a cyclic quadrilateral with sides a,b,c,d, and diagonals e,f:
Try this Drag any orange dot. Note how the diagonals formula always holds.
Recall that an
inscribed (or 'cyclic') quadrilateral is one where the four
vertices all lie on a circle.
It turns out that there is a relationship between the sides of the quadrilateral and its diagonals.
If you mutiply the lengths of each pair of opposite sides, the sum of these products equals the product of the diagonals.
Formally: For a cyclic quadrilateral with sides a,b,c,d and diagonals e,f, then
In the figure above, drag any vertex around the circle. Note how the relationship holds.
* NOTE The lengths in the figure above are rounded to one decimal place for clarity. If you repeat the calculation on a calculator you may not therefore get an exact equality. The equation is meant only to illustrate how the theorem works.
'Crossed' polygons
In the figure above, if you drag a point past its neighbor the quadrilateral will become 'crossed' where one side crossed over another.
In such 'crossed' quadrilaterals Ptolemy's Theorem no longer holds.
(Most properties of polygons are invalid when the polygon is crossed).
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
(C) 2011 Copyright Math Open Reference. All rights reserved
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