Reuschle's theorem

Reuschle's theorem

The set of Reuschle found and in 1853 published by the German scholar Karl Gustav Reuschle , is a theorem of elementary Euclidean geometry and as such located between triangle and circle geometry . It is sometimes referred to as Terquem's theorem, after the French mathematician Olry Terquem , who published the theorem in 1842. The sentence deals with a question about the intersection point properties of certain corner transversals , which one encounters in a similar form in connection with the Euler straight line and the Feuerbach nine-point circle . The proof of Reuschle's theorem is based on the secant theorem as well as the theorem of Ceva and its inverse theorem .

Formulation of the sentence

The sentence can be stated in modern terms as follows:

Let there be a triangle in the Euclidean plane and a circle , which should cut a chord from each side of the triangle . ${\displaystyle ABC}$ ${\displaystyle {\mathcal {K}}}$

It should be the cornerstone in the opposite side of the triangle contained chord the route so . ${\displaystyle X\in \{A,B,C\}}$ ${\displaystyle {\overline {{X_{1}}{X_{2}}}}\;(X_{1},X_{2}\in {\mathcal {K}}\;,\;X_{1}\neq X_{2})}$ ${\displaystyle {\overline {{A_{1}}{A_{2}}}}\subsetneqq {\overline {BC}}\;,\;{\overline {{B_{1}}{B_{2}}}}\subsetneqq {\overline {AC}}\;,\;{\overline {{C_{1}}{C_{2}}}}\subsetneqq {\overline {AB}}}$

Each vertex will be with the two opposite tendon end points by the associated Ecktransversalen connected . ${\displaystyle X}$ ${\displaystyle X_{1},X_{2}}$ ${\displaystyle {XX_{1}},{XX_{2}}}$

Then:

If the first three corner transversals meet at a common point of intersection , the other three corner transversals also meet at a common point of intersection . ${\displaystyle {AA_{1}},{BB_{1}},{CC_{1}}}$ ${\displaystyle S_{1}}$ ${\displaystyle {AA_{2}},{BB_{2}},{CC_{2}}}$ ${\displaystyle S_{2}}$

In other words:

If one places the three associated corner transversals with the base points in a triangle of the Euclidean plane through a given inner point and the circumference of the base triangle cuts three circular chords from the sides of the triangle , the corner transversals thus given also have a common point of intersection . ${\displaystyle ABC}$ ${\displaystyle S_{1}\in (\operatorname {conv} \{A,B,C\})^{\circ }}$ ${\displaystyle A_{1}\in {\overline {BC}},B_{1}\in {\overline {AC}},C_{1}\in {\overline {AB}}}$ ${\displaystyle A_{1}B_{1}C_{1}}$ ${\displaystyle {\overline {{A_{1}}{A_{2}}}}\subsetneqq {\overline {BC}}\;,\;{\overline {{B_{1}}{B_{2}}}}\subsetneqq {\overline {AC}}\;,\;{\overline {{C_{1}}{C_{2}}}}\subsetneqq {\overline {AB}}}$ ${\displaystyle {AA_{2}},{BB_{2}},{CC_{2}}}$ ${\displaystyle S_{2}}$

literature

• Friedrich Joseph Pythagoras Riecke (Hrsg.): Mathematische Unterhaltungen . First issue. Dr. Martin Sendet , Walluf near Wiesbaden 1973, ISBN 3-500-26010-1 (unchanged reprint of the Stuttgart edition 1867–1873).

Web links

Commons : Reuschle's set  - collection of images, videos and audio files

• Terquem's theorem on cut-the-knot.org
• Eric W. Weisstein : Cyclocevian Conjugate . In: MathWorld (English).

Individual evidence

1. ↑ Friedrich Joseph Pythagoras Riecke (Ed.): Mathematische Unterhaltungen. First issue. 1973, p. 125