Tendon quadrangle

A chordal quadrilateral ABCD with circumference k

A chordal quadrilateral is a quadrilateral whose corner points lie on a circle , the circumference of the quadrilateral. Hence all sides of the quadrilateral tendons are chords of the circumference. Usually one means with tendon quadrilateral a non-crossed tendon quadrilateral, this is necessarily convex .

The isosceles trapezoid and the rectangle are special quadrilateral tendons.

sentences

The set of chords applies to every quadrilateral tendon :

• The products of two opposite diagonal sections are the same size. That is, if the intersection of the two diagonals is and , then applies . ${\displaystyle P}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle {\overline {BD}}}$ ${\displaystyle {\overline {AP}}\cdot {\overline {CP}}={\overline {BP}}\cdot {\overline {DP}}}$

The following sentences only apply to chord quadrilaterals ABCD that have not been crossed over :

• Opposite angles add up to 180 °, ie . ${\displaystyle \alpha +\gamma =\beta +\delta =180^{\circ }}$
• Ptolemy's theorem : The sum of the products of opposite sides of the tendon square is equal to the product of the diagonals: . ${\displaystyle {\overline {AB}}\cdot {\overline {CD}}+{\overline {BC}}\cdot {\overline {DA}}={\overline {AC}}\cdot {\overline {BD}}}$

properties

In the chordal quadrilateral the sum of the angles of the opposite angles is 180 °.

${\displaystyle \alpha +\gamma =180^{\circ }}$

${\displaystyle \beta +\delta =180^{\circ }}$

The proof results directly from the theorem of circular angles , since two opposing angles of the chordal quadrilateral are circumferential angles over two complementary circular arcs whose central angles add up to 360 °. Since circumferential angles are half the size of the central angle over the same arc, the circumferential angles must add up to 360 ° / 2 = 180 °.

Another evidence can be found in the evidence archive .

The reverse of this statement is also true; H. if the sum of the opposite angles is 180 ° in a square , then it is a chordal square .

Formulas

Mathematical formulas for the chord square
Area	${\displaystyle A\,=\,{\sqrt {(s-a)(s-b)(s-c)(s-d)}}}$ With ${\displaystyle s={\frac {a+b+c+d}{2}}}$
	${\displaystyle A={\frac {e\cdot (a\cdot b+c\cdot d)}{4\cdot R}}={\frac {f\cdot (a\cdot d+b\cdot c)}{4\cdot R}}}$
Length of the diagonal	${\displaystyle e={\sqrt {\frac {(a\cdot c+b\cdot d)\cdot (a\cdot d+b\cdot c)}{a\cdot b+c\cdot d}}}}$
	${\displaystyle f={\sqrt {\frac {(a\cdot b+c\cdot d)\cdot (a\cdot c+b\cdot d)}{a\cdot d+b\cdot c}}}}$
Perimeter radius	${\displaystyle R={\frac {1}{4\cdot A}}\cdot {\sqrt {(a\cdot b+c\cdot d)\cdot (a\cdot c+b\cdot d)\cdot (a\cdot d+b\cdot c)}}}$
Interior angle	${\displaystyle \alpha +\gamma =\beta +\delta =180^{\circ }}$

The first formula for area is a generalization of Heron's theorem for triangles and is also known as the Brahmagupta theorem or Brahmagupta's formula . Here one understands a triangle as a degenerate chordal quadrilateral whose fourth side has the length 0, i.e. H. two of its corner points lie on top of one another. Brahmagupta's formula can be generalized to Bretschneider's formula , which adds a correction term to Brahmagupta's formula, which is 0 in the case of a quadrilateral , and then applies to any quadrilateral .

A square with fixed, ordered side lengths has the largest possible surface area if and only if it is a chordal square . Likewise, a polygon has the largest area if and only if it is a chordal polygon .

Equations

The following equations apply to the interior angles of a chordal quadrilateral :

${\displaystyle \sin(\alpha )={\frac {2\cdot {\sqrt {(s-a)\cdot (s-b)\cdot (s-c)\cdot (s-d)}}}{a\cdot d+b\cdot c}}}$

${\displaystyle \cos(\alpha )={\frac {a^{2}+d^{2}-b^{2}-c^{2}}{2\cdot (a\cdot d+b\cdot c)}}}$

${\displaystyle \tan \left({\frac {\alpha }{2}}\right)={\sqrt {\frac {(s-a)\cdot (s-d)}{(s-b)\cdot (s-c)}}}}$

The following applies to the angle of intersection of the diagonals :

${\displaystyle \tan \left({\frac {\theta }{2}}\right)={\sqrt {\frac {(s-b)\cdot (s-d)}{(s-a)\cdot (s-c)}}}}$

The following applies to the angle of intersection of the sides a and c:

${\displaystyle \cos \left({\frac {\varphi }{2}}\right)={\sqrt {\frac {(s-b)\cdot (s-d)\cdot (b+d)^{2}}{(a\cdot b+c\cdot d)\cdot (a\cdot d+b\cdot c)}}}}$

See also

• Japanese phrase for square tendons
• Tangent square
• Tendon polygon

literature

• IN Bronstein , KA Semendjajev , G. Musiol, H. Mühlig (Hrsg.): Taschenbuch der Mathematik . 7th, completely revised and expanded edition. Verlag Harri Deutsch, Frankfurt am Main 2008, ISBN 978-3-8171-2007-9 .
• H. Fenkner, K. Holzmüller: Mathematical teaching work. According to the guidelines for the curricula of the higher schools in Prussia, revised by Dr. Karl Holzmüller . Geometry. Edition A in 2 parts. 12th edition. I. part. Published by Otto Salle, Berlin 1926.
• Theophil Lambacher , Wilhelm Schweizer (Ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools . Geometry. Edition E. Part 1. 15th edition. Ernst Klett Verlag, Stuttgart 1965.
• Harald Scheid (Ed.): DUDEN: Rechnen und Mathematik . 4th, completely revised edition. Bibliographical Institute, Mannheim / Vienna / Zurich 1985, ISBN 3-411-02423-2 .
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . 5th volume: Sed to Zyl. Spectrum Academic Publishing House, Heidelberg / Berlin 2002, ISBN 3-8274-0437-1 .

Web links

Wiktionary: Sehnenviereck  - explanations of meanings, word origins, synonyms, translations

Wikibooks: Proof of Ptolemy's Theorem  - Learning and Teaching Materials

Wikibooks: Proof of the size of the opposite angle in the tendon quadrangle  - learning and teaching materials

• Proof of Ptolemy's theorem (with inversion) with geometric means of the grammar school middle level, state education server Baden-Württemberg

Individual evidence

1. ↑ Titu Andreescu, Oleg Mushkarov, Luchezar N. Stoyanov: Geometric problem on maxima and minima. Birkhäuser, Boston a. a. 2006, ISBN 0-8176-3517-3 , p. 69 ( excerpt (Google) ).
2. ^ CV Durell, A. Robson: Advanced Trigonometry