String set

String set

The chord set is a set from elementary geometry and describes a relationship between the lines that are formed by two intersecting circular chords .

More precisely it says:

If two chords intersect at one point in a circle , the product of the sections formed thereby on one chord is equal to the product of the sections on the other chord. ${\displaystyle S}$

Formulation of the sentence

Consider a circle with two chords that intersect at one point . The points of intersection of the circle with one chord are marked with and , those with the other chord with and . Then: ${\displaystyle S}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle D}$

${\displaystyle {\overline {AS}}\cdot {\overline {CS}}={\overline {BS}}\cdot {\overline {DS}}}$

The statement can also be formulated as a ratio equation:

${\displaystyle {\overline {AS}}:{\overline {DS}}={\overline {BS}}:{\overline {CS}}}$

reversal

The reverse of the proposition also applies: If the following applies to the diagonals of a square with the diagonal intersection: ${\displaystyle ABCD}$ ${\displaystyle S}$

${\displaystyle {\overline {AS}}\cdot {\overline {CS}}={\overline {BS}}\cdot {\overline {DS}}}$

then this square has a circumference , that is, it is a chordal square .

Relation to the rate of altitude

String set as height set
| CD || DE | = | AD || DB | <=> h ² = pq

The chord theorem can also be understood as a generalization of Euclid's theorem of heights . If one chooses the two tendons so that one of them corresponds to the diameter and the other is perpendicular to it, their endpoints with the endpoints of the diameter form a right triangle according to Thales's theorem and the statement of the chordal set corresponds in this configuration to of Euclid's theorem of heights.

Derivation

${\displaystyle \triangle ASD\sim \triangle BSC}$

The set results directly from similar triangles occurring in the configuration . The following applies to the triangles ASD and BSC :

${\displaystyle {\begin{aligned}\angle ADS&=\angle SCB\,({\text{Umfangswinkel ueber AB}})\\\angle SAD&=\angle CBS\,({\text{Umfangswinkel ueber CD}})\\\angle DSA&=\angle BSC\,({\text{Scheitelwinkel}})\end{aligned}}}$

So the two triangles are similar and it follows:

${\displaystyle {\frac {AS}{SD}}={\frac {BS}{SC}}\Leftrightarrow |AS|\cdot |SC|=|BS|\cdot |SD|}$

See also

• Secant set
• Secant-Tangent Theorem
• Potency (geometry) , combines the statements of the chord, secant and secant-tangent set in a uniform concept

literature

• Max Koecher , Aloys Krieg : level geometry. 2nd revised and expanded edition. Springer-Verlag, Berlin a. a. 2000, ISBN 3-540-67643-0 ( Springer textbook ), p. 148
• Hans Schupp: Elementary Geometry. Schöningh, Paderborn 1977, ISBN 3-506-99189-2 , p. 149 ( Uni-Taschenbücher 669).
• Schülerduden - Mathematics I . Bibliographisches Institut & FA Brockhaus, 8th edition, Mannheim 2008, ISBN 978-3-411-04208-1 , pp. 415-417

Web links

Wikibooks: Proof of the Chord Set  - Learning and Teaching Materials

• Intersecting Chords Theorem on cut-the-knot.org
• Intersecting Chords Theorem on proofwiki.org
• Eric W. Weisstein : Chord . In: MathWorld (English).