Ptolemy's Theorem

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 multiply 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.

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).

- Polygon general definition
- Quadrilateral
- Regular polygon
- Irregular polygon
- Convex polygons
- Concave polygons
- Polygon diagonals
- Polygon triangles
- Apothem of a regular polygon
- Polygon center
- Radius of a regular polygon
- Incircle of a regular polygon
- Incenter of a regular polygon
- Circumcircle of a polygon
- Parallelogram inscribed in a quadrilateral

- Square
- Diagonals of a square
- Rectangle
- Diagonals of a rectangle
- Golden rectangle
- Parallelogram
- Rhombus
- Trapezoid
- Trapezoid median
- Kite
- Inscribed (cyclic) quadrilateral

- Regular polygon area
- Irregular polygon area
- Rhombus area
- Kite area
- Rectangle area
- Area of a square
- Trapezoid area
- Parallelogram area

- Perimeter of a polygon (regular and irregular)
- Perimeter of a triangle
- Perimeter of a rectangle
- Perimeter of a square
- Perimeter of a parallelogram
- Perimeter of a rhombus
- Perimeter of a trapezoid
- Perimeter of a kite

- Exterior angles of a polygon
- Interior angles of a polygon
- Relationship of interior/exterior angles
- Polygon central angle

- Tetragon, 4 sides
- Pentagon, 5 sides
- Hexagon, 6 sides
- Heptagon, 7 sides
- Octagon, 8 sides
- Nonagon Enneagon, 9 sides
- Decagon, 10 sides
- Undecagon, 11 sides
- Dodecagon, 12 sides

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