Ptolemy's theorem

In mathematics, Ptolemy's theorem is a relation in Euclidean geometry between the four sides and two diagonals or chords of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus).

If the quadrilateral is given by its four vertices "A", "B", "C", and "D" in order, then the theorem states that:

: $overline\{AC\}cdot\; overline\{BD\}=overline\{AB\}cdot\; overline\{CD\}+overline\{BC\}cdot\; overline\{AD\}$

where the overbar denotes the lengths of the line segments between the named vertices.

This relation may be verbally expressed as follows:

:"If a quadrilateral is inscribed in a circle then the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals."

Moreover, the converse of Ptolemy's theorem is also true:

:"In a quadrilateral, if the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals, then the quadrilateral can be inscribed in a circle."

Examples

* Any square can be inscribed in a circle whose center is the barycenter of the square. If the common length of its four sides is equal to $a$ then the length of the diagonal is equal to $sqrt\{2\}a$ according to the Pythagorean theorem and the relation obviously holds.
* More generally, if the quadrilateral is a rectangle with sides a and b and diagonal c then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then c², the right hand side of Ptolemy's relation is the sum "a"² + "b"².
*A more interesting example is the relation between the length "a" of the side and the (common) length "b" of the 5 chords in a regular pentagon. In this case the relation reads "b"² = "a"² + "ab" which yields the golden ratio ::$varphi\; =\; \{b\; over\; a\}\; =\; over\{overline\{P\_1P\_4\}cdot\; overline\{P\_2P\_3\}\; =1+$overline{P_1P_2}cdot overline{P_3P_4over{overline{P_1P_4}cdot overline{P_2P_3}

can now be interpreted as the algebraic relation (already used above) between cross-ratios

:$$(z_1-z_3)(z_2-z_4)}over{(z_1-z_4)(z_2-z_3)=1-(z_1-z_2)(z_3-z_4)}over{(z_1-z_4)(z_3-z_2)

using the representation of the vertices $P\_1,ldots,P\_4$ as the points $z\_1,\; ldots,\; z\_4$ on the unit circle.

Corollaries

In the case of a circle of unit diameter the sides $S\_1,S\_2,S\_3,S\_4;$ of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles $heta\_1,\; heta\_2,\; heta\_3,$ and $heta\_4,$ which they subtend. Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend. We may then write Ptolemy's Theorem in the following trigonometric form:

:$sin\; heta\_1sin\; heta\_3+sin\; heta\_2sin\; heta\_4=sin(\; heta\_3+\; heta\_2)sin(\; heta\_3+\; heta\_4);$

Applying certain conditions to the subtended angles $heta\_1,\; heta\_2,\; heta\_3,$ and $heta\_4,$ it is possible to derive a number of important corollaries using the above as our starting point. In what follows it is important to bear in mind that the sum of angles $heta\_1+\; heta\_2+\; heta\_3+\; heta\_4=180^circ,$.

Corollary 1. Pythagoras' theorem

Let $heta\_1=\; heta\_3;$ and $heta\_2=\; heta\_4;$ $Rightarrow\; heta\_1+\; heta\_2=\; heta\_3+\; heta\_4=90^circ;$(Since opposite angles of a cyclic quadrilateral are supplementary). Then: [In De Revolutionibus Orbium Coelestium, Copernicus does not refer to Pythagoras' Theorem by name but uses the term 'Porism' – a word which in this particular context would appear to denote an observation on – or obvious consequence of – another existing theorem. The 'Porism' can be viewed on pages [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1543droc.book.....C&db_key=AST&page_ind=36&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES 36 and 37] of DROC (Harvard electronic copy)]

:

Corollary 2. The law of cosines

Let $heta\_2=\; heta\_4;$. The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. The parallel sides differ in length by 2x units where::$x=S\_2.cos(\; heta\_2+\; heta\_3);$

It will be easier in this case to revert to the standard statement of Ptolemy's theorem:

:

The cosine rule for triangle ABC.

Corollary 3: Compound angle sine (+)

Let $heta\_1+\; heta\_2=\; heta\_3+\; heta\_4=90^circ;$

:

Formula for compound angle sine (+)cite web | url = http://www.cut-the-knot.org/proofs/sine_cosine.shtml | title = Sine, Cosine, and Ptolemy's Theorem ]

Corollary 4: Compound angle sine (-)

Let $heta\_1=90^circ\; Rightarrow\; heta\_2+(\; heta\_3+\; heta\_4)=90^circ$

:

Formula for compound angle sine (-).

This derivation corresponds to the [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1543droc.book.....C&db_key=AST&page_ind=38&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES Third Theorem] as chronicled by Copernicus following Ptolemy in Almagest. In particular if the sides of a pentagon (subtending 36° at the circumference) and of a hexagon (subtending 30° at the circumference) are given, a chord subtending 6° may be calculated. This was a critical step in the ancient method of calculating tables of chords. [To understand the Third Theorem, compare the Copernican diagram shown on page 39 of the [http://ads.harvard.edu/books/1543droc.book/ Harvard copy] of De Revolutionibus to that for the derivation of sin(A-B) found in the above [http://www.cut-the-knot.org/proofs/sine_cosine.shtml cut-the-knot] web page]

Corollary 5: Compound angle cosine (+)

This corollary is the core of the [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1543droc.book.....C&db_key=AST&page_ind=39&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES Fifth Theorem] as chronicled by Copernicus following Ptolemy in Almagest.

Let $heta\_3=90^circ\; Rightarrow\; heta\_1+(\; heta\_2+\; heta\_4)=90^circ$

:

Formula for compound angle cosine (+)

Despite lacking the dexterity of our modern trigonometric notation, it should be clear from the above corollaries that in Ptolemy's theorem (or more simply the [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1543droc.book.....C&db_key=AST&page_ind=37&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES Second Theorem] ) the ancient world had at its disposal an extremely flexible and powerful trigonometric tool which enabled the cogniscenti of those times to draw up accurate tables of chords (corresponding to tables of sines) and to use these in their attempts to understand and map the cosmos as they saw it. Since tables of chords were drawn up by Hipparchus three centuries before Ptolemy, we must assume he knew of the 'Second Theorem' and its derivatives. Following the trail of ancient astronomers, history records the star catalogue of Timocharis of Alexandria. If, as seems likely, the compilation of such catalogues required an understanding of the 'Second Theorem' then the true origins of the latter disappear thereafter into the mists of antiquity but it cannot be unreasonable to presume that the astronomers, architects and construction engineers of ancient Egypt may have had some knowledge of it.

Ptolemy's inequality

The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. It states that, given a quadrilateral "ABCD", then

: $overline\{AB\}cdot\; overline\{CD\}+overline\{BC\}cdot\; overline\{DA\}\; ge\; overline\{AC\}cdot\; overline\{BD\}$

where equality holds if and only if the quadrilateral is cyclic. This special case is equivalent to Ptolemy's theorem.

See also

* Golden ratio
* Casey's theorem

References

De Revolutionibus Orbium Coelestium, Copernicus, Nicolaus. English translation from On the Shoulders of Giants, Hawking, S 2002, Penguin Books. ISBN 0-141-01571-3

Footnotes

External links

* [http://www.mathpages.com/home/kmath099.htm MathPages - On Ptolemy's Theorem]
*
* [http://www.cut-the-knot.org/proofs/ptolemy.shtml Ptolemy's Theorem] at cut-the-knot
* [http://www.cut-the-knot.org/proofs/sine_cosine.shtml Compound angle proof] at cut-the-knot
* [http://planetmath.org/encyclopedia/PtolemysTheorem.html Ptolemy's Theorem] on PlanetMath
* [http://mathworld.wolfram.com/PtolemyInequality.html Ptolemy Inequality] on MathWorld
* [http://ads.harvard.edu/books/1543droc.book/ De Revolutionibus Orbium Coelestium] at Harvard.
* [http://www.atara.net/deep_secrets/index.html Deep Secrets: The Great Pyramid, the Golden Ratio and the Royal Cubit]
*" [http://demonstrations.wolfram.com/PtolemysTheorem/ Ptolemy's Theorem] " by Jay Warendorff, The Wolfram Demonstrations Project.
* [http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/bookXIII.html Book XIII] of [http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html Euclid's Elements]