Secant-Tangent Theorem

{\ displaystyle \ Rightarrow}

{\ displaystyle \ angle PG_ {2} T = \ angle PTG_ {1}}

{\ displaystyle \ Rightarrow}

{\ displaystyle \ triangle PTG_ {2} \ sim \ triangle PG_ {1} T}

{\ displaystyle \ Rightarrow}

{\ displaystyle {\ frac {| PT |} {| PG_ {2} |}} = {\ frac {| PG_ {1} |} {| PT |}}}

{\ displaystyle \ Rightarrow}

{\ displaystyle | PT | ^ {2} = | PG_ {1} | \ cdot | PG_ {2} |}

The secant-tangent theorem (also called chord-tangent theorem ) is a theorem of Euclidean geometry . It describes the relationship between lines formed by a tangent, a secant and the associated circle (see drawing).

Given is a circle with a secant and a tangent that intersect at a point outside the circle. If the points of intersection of the circle are designated as respectively and the point of contact of the tangent as , then the following applies: ${\ displaystyle g}$ ${\ displaystyle t}$ ${\ displaystyle P}$ ${\ displaystyle g}$ ${\ displaystyle G_ {1}}$ ${\ displaystyle G_ {2}}$ ${\ displaystyle T}$

{\ displaystyle {\ overline {PG_ {1}}} \ cdot {\ overline {PG_ {2}}} = {\ overline {PT}} ^ {2}}

This statement can also be formulated as a ratio equation:

{\ displaystyle {\ overline {PG_ {1}}}: {\ overline {PT}} = {\ overline {PT}}: {\ overline {PG_ {2}}}}

The secant-tangent theorem describes a special case of the secant theorem in which the intersection points of the second secant and the circle coincide at one point. The propositions can be proven - similarly to the string proposition - with the help of similar triangles . All three sentences can be summarized with the help of the concept of potency or unified into a single statement.

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

Circular sets - teaching materials from the University of Lüneburg

Emese Vargyas, Ysette Weiss-Pidstrygach: Which areas is the set of tendons about? - Slides for a lecture

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