Tangent square

A tangent square ABCD with an incircle k

A tangent square is a square whose sides are tangents of a circle . This circle is called the inscribed circle of the tangent square. Such a tangent square is always convex. Quadrilaterals in which only the extended sides are tangents of a circle and which therefore do not necessarily have to be convex are not tangent quadrilaterals in the sense of the definition here. Special tangent quadrilaterals are the square , the diamond and the dragon quadrangle .

properties

For every tangent square, Pitot's theorem applies : The sum of two opposite sides is equal to the sum of the other two sides. So it applies

${\displaystyle a+c=b+d}$

Conversely, it is also true that every convex square has an inscribed circle and is therefore a tangent square . Pitot's theorem and its inverse are collectively referred to as the tangent quadrilateral theorem. ${\displaystyle a+c=b+d}$

The center of the inscribed circle is the intersection of the bisector of all four interior angles . For this reason, all the bisectors of the tangent square must also intersect at one point .

In addition, a square that is not a trapezoid is a tangent square if and only if one of the following conditions applies:

• ${\displaystyle BE+BF=DE+DF}$
• ${\displaystyle AE-EC=AF-FC}$

E is the intersection of the straight lines and and F is the intersection of the straight lines and . ${\displaystyle AB}$ ${\displaystyle CD}$ ${\displaystyle BC}$ ${\displaystyle DA}$

Formulas

Mathematical formulas for the tangent square
Area	${\displaystyle A=r\cdot (a+c)=r\cdot (b+d)}$
	${\displaystyle A={\frac {1}{2}}\cdot {\sqrt {p^{2}\cdot q^{2}-(a\cdot c-b\cdot d)^{2}}}}$
	${\displaystyle A={\sqrt {(e+f+g+h)\cdot (e\cdot f\cdot g+f\cdot g\cdot h+g\cdot h\cdot e+h\cdot e\cdot f)}}}$
	${\displaystyle A={\sqrt {a\cdot b\cdot c\cdot d-(e\cdot g-f\cdot h)^{2}}}}$
	${\displaystyle A={\sqrt {a\cdot b\cdot c\cdot d}}\cdot \sin \left({\frac {\alpha +\gamma }{2}}\right)={\sqrt {a\cdot b\cdot c\cdot d}}\cdot \sin \left({\frac {\beta +\delta }{2}}\right)}$
scope	${\displaystyle U=2\cdot (a+c)=2\cdot (b+d)}$
Length of the diagonal	${\displaystyle p={\sqrt {\frac {(e+g)\cdot ((e+g)\cdot (f+h)+4\cdot f\cdot h)}{f+h}}}}$
	${\displaystyle q={\sqrt {\frac {(f+h)\cdot ((e+g)\cdot (f+h)+4\cdot e\cdot g)}{e+g}}}}$
Inscribed radius	${\displaystyle r={\frac {A}{a+c}}={\frac {A}{b+d}}}$
	${\displaystyle r={\sqrt {\frac {e\cdot f\cdot g+f\cdot g\cdot h+g\cdot h\cdot e+h\cdot e\cdot f}{e+f+g+h}}}}$

An interesting special case is when a tangent quadrilateral satisfies the condition

${\displaystyle \alpha +\gamma =\beta +\delta }$

Fulfills. Under this condition, the tangential quadrilateral is also a quadrilateral , so a square with inscribed circle and a radius . In this case, the formula for the area of quadrilateral tendons provides the simple result

${\displaystyle A={\sqrt {a\cdot b\cdot c\cdot d}}}$

Equations

The following equations apply to the angles of each tangent quadrilateral :

${\displaystyle \sin \left({\frac {\alpha }{2}}\right)={\sqrt {\frac {e\cdot f\cdot g+f\cdot g\cdot h+g\cdot h\cdot e+h\cdot e\cdot f}{(e+f)\cdot (e+g)\cdot (e+h)}}}}$

${\displaystyle \sin \left({\frac {\beta }{2}}\right)={\sqrt {\frac {e\cdot f\cdot g+f\cdot g\cdot h+g\cdot h\cdot e+h\cdot e\cdot f}{(f+e)\cdot (f+g)\cdot (f+h)}}}}$

${\displaystyle \sin \left({\frac {\gamma }{2}}\right)={\sqrt {\frac {e\cdot f\cdot g+f\cdot g\cdot h+g\cdot h\cdot e+h\cdot e\cdot f}{(g+e)\cdot (g+f)\cdot (g+h)}}}}$

${\displaystyle \sin \left({\frac {\delta }{2}}\right)={\sqrt {\frac {e\cdot f\cdot g+f\cdot g\cdot h+g\cdot h\cdot e+h\cdot e\cdot f}{(h+e)\cdot (h+f)\cdot (h+g)}}}}$

${\displaystyle \tan \left({\frac {\angle ABD}{2}}\right)\cdot \tan \left({\frac {\angle BDC}{2}}\right)=\tan \left({\frac {\angle ADB}{2}}\right)\cdot \tan \left({\frac {\angle DBC}{2}}\right)}$

See also

• Tendon quadrangle
• Tangent polygon

literature

• Hartmut Wellstein, Peter Kirsche: Elementary Geometry . Springer, 2009, ISBN 978-3-8348-0856-1 , pp. 60-61
• Siegfried Krauter, Christine Bescherer: The Elementary Geometry Experience: A workbook for independent and active discovery . Springer, 2012, ISBN 978-3-8274-3025-0 , pp. 77-78
• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: With harmonious proportions to conic sections: pearls of classical geometry . Springer 2016, ISBN 978-3-662-53034-4 , p. 21; Extract (PDF)

Web links

Commons : Tangentenviereck  - collection of images, videos and audio files

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

• Eric W. Weisstein : Tangential Quadrilateral . In: MathWorld (English).

Individual evidence

1. ^ Martin Josefsson: Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral , Forum Geometricorum
2. ↑ Nicusor Minculete: Characterizations of a Tangential Quadrilateral , Forum Geometricorum