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Difference between revisions of "Bodenmiller theorem"

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Consider a complete [[Quadrilateral|quadrilateral]] with four sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106804.png" /> intersecting in the six points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b1106809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b11068010.png" />. Then the three circles that have the three diagonals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b11068011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b11068012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110680/b11068013.png" /> as diagonals intersect in the same two points.
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Consider a complete [[Quadrilateral|quadrilateral]] with four sides $a$, $b$, $c$, $d$ intersecting in the six points $P$, $Q$, $R$, $S$, $T$, $U$. Then the three circles that have the three diagonals $PS$, $QT$ and $RU$ as diagonals intersect in the same two points.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b110680a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b110680a.gif" />

Revision as of 10:09, 12 April 2014

Consider a complete quadrilateral with four sides $a$, $b$, $c$, $d$ intersecting in the six points $P$, $Q$, $R$, $S$, $T$, $U$. Then the three circles that have the three diagonals $PS$, $QT$ and $RU$ as diagonals intersect in the same two points.

Figure: b110680a

References

[a1] R. Fritsch, "Remarks on Bodenmiller's theorem" J. of Geometry , 47 (1993) pp. 23–31
[a2] G. Weiss, "Eine räumliche Deutung der Vierseiteigenschaft von Bodenmiller" Elemente der Math. , 37 (1982) pp. 21–23
How to Cite This Entry:
Bodenmiller theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bodenmiller_theorem&oldid=31611
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article