Secant set

The secant theorem says: If two secants outside the circle intersect at one point , the product of the section lengths from the secant intersection to the two intersections of the circle and secant is the same on both secants. Shorter: The product of the secant segments is constant. ${\displaystyle P}$

Secant set

Formulation of the sentence

Consider a circle with two secants that intersect at a point outside the circle. If the points of intersection of the circle with one secant are designated as and and the points of intersection with the other secant as and , then: ${\displaystyle P}$ ${\displaystyle A}$ ${\displaystyle D}$ ${\displaystyle B}$ ${\displaystyle C}$

${\displaystyle {\overline {AP}}\cdot {\overline {DP}}={\overline {BP}}\cdot {\overline {CP}}}$

This statement can also be formulated as a ratio equation:

${\displaystyle {\overline {AP}}:{\overline {BP}}={\overline {CP}}:{\overline {DP}}}$

Proof idea

The secant theorem can be proven - similar to the chord theorem and the secant-tangent theorem - with the help of similar triangles .

The triangles and are similar triangles because: ${\displaystyle APC}$ ${\displaystyle BPD}$

1. The angle in point is common to both triangles. ${\displaystyle \varphi }$ ${\displaystyle P}$
2. Circumferential angles over a chord are the same. Applying this theorem to the tendon gives . ${\displaystyle [AB]}$ ${\displaystyle \angle ADB=\angle ACB}$

${\displaystyle \triangle APC\sim \triangle BPD}$( Similarity theorem WW )

This results in the relationship equation

${\displaystyle {\overline {AP}}:{\overline {BP}}={\overline {CP}}:{\overline {DP}}}$.

Multiplication with gives: ${\displaystyle {\overline {BP}}\cdot {\overline {DP}}}$

${\displaystyle {\overline {AP}}\cdot {\overline {DP}}={\overline {BP}}\cdot {\overline {CP}}}$

See also

• String set
• Secant-Tangent Theorem
• Power (geometry) , combines the statement of vision, secent and secant tangent theorem in a uniform concept

literature

• Max Koecher , Aloys Krieg: level geometry . 2nd Edition. Springer-Verlag Berlin Heidelberg New York, 2000, ISBN 3-540-67643-0 , p. 148
• H. Schupp: Elementargeometrie , UTB Schöningh (1977), ISBN 3-506-99189-2 , p. 150
• 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 Secant Theorem  - Learning and Teaching Materials

• Power of a Point Theorem on cut-the-knot.org