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Difference between revisions of "Nine-point circle"

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''Euler circle''
 
''Euler circle''
  
A circle whose periphery contains the midpoints of the sides of a triangle, the bases of its altitudes, and the midpoints of the segment connecting the orthocentre of the triangle with the vertices. Its radius is equal to one-half of the radius of the circle circumscribed about the triangle. The nine-point circle of a triangle is tangent to the circle inscribed in it and to the three escribed circles. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667501.png" /> be the orthocentre of a non-equilateral triangle, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667502.png" /> be the centre of gravity, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667503.png" /> be the centre of the circumscribed circle and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667504.png" /> be the centre of the nine-point circle. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667505.png" /> then lie on a straight line (Euler's line), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667506.png" /> being the midpoint of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667507.png" />, and the pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667508.png" /> harmonically subdivides the pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n0667509.png" />.
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A circle whose periphery contains the midpoints of the sides of a triangle, the bases of its altitudes, and the midpoints of the segment connecting the orthocentre of the triangle with the vertices. Its radius is equal to one-half of the radius of the circle circumscribed about the triangle. The nine-point circle of a triangle is tangent to the circle inscribed in it and to the three escribed circles. Let $H$ be the orthocentre of a non-equilateral triangle, let $T$ be the centre of gravity, let $O$ be the centre of the circumscribed circle and let $E$ be the centre of the nine-point circle. The points $H,T,O,E$ then lie on a straight line (Euler's line), $E$ being the midpoint of the segment $HO$, and the pair of points $H,T$ harmonically subdivides the pair of points $O,E$.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066750a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066750a.gif" />
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Sometimes the nine-point circle is referred to as the Feuerbach circle. The fact that the nine-point circle is tangent to the inscribed circle and the three escribed circles is Feuerbach's theorem.
 
Sometimes the nine-point circle is referred to as the Feuerbach circle. The fact that the nine-point circle is tangent to the inscribed circle and the three escribed circles is Feuerbach's theorem.
  
More generally one has the nine-point conic and the eleven-point conic determined by a projective base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066750/n06675010.png" /> (giving a coordinate system) in the projective plane, cf. [[#References|[a2]]], Sects. 16.5.5.1, 16.7.5.
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More generally one has the nine-point conic and the eleven-point conic determined by a projective base $\{a,b,c,d\}$ (giving a coordinate system) in the projective plane, cf. [[#References|[a2]]], Sects. 16.5.5.1, 16.7.5.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  pp. Sects. 10.11.3, 17.5.4  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Veblen,  J.W. Young,  "Projective geometry" , '''II''' , Blaisdell  (1946)  pp. 169; 233</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  pp. Sects. 10.11.3, 17.5.4  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Veblen,  J.W. Young,  "Projective geometry" , '''II''' , Blaisdell  (1946)  pp. 169; 233</TD></TR></table>

Revision as of 10:43, 16 April 2014

Euler circle

A circle whose periphery contains the midpoints of the sides of a triangle, the bases of its altitudes, and the midpoints of the segment connecting the orthocentre of the triangle with the vertices. Its radius is equal to one-half of the radius of the circle circumscribed about the triangle. The nine-point circle of a triangle is tangent to the circle inscribed in it and to the three escribed circles. Let $H$ be the orthocentre of a non-equilateral triangle, let $T$ be the centre of gravity, let $O$ be the centre of the circumscribed circle and let $E$ be the centre of the nine-point circle. The points $H,T,O,E$ then lie on a straight line (Euler's line), $E$ being the midpoint of the segment $HO$, and the pair of points $H,T$ harmonically subdivides the pair of points $O,E$.

Figure: n066750a

References

[1] S.I. Zetel', "A new geometry of triangles" , Moscow (1962) (In Russian)
[2] D.I. Perepelkin, "A course of elementary geometry" , 1 , Moscow-Leningrad (1948) (In Russian)


Comments

Sometimes the nine-point circle is referred to as the Feuerbach circle. The fact that the nine-point circle is tangent to the inscribed circle and the three escribed circles is Feuerbach's theorem.

More generally one has the nine-point conic and the eleven-point conic determined by a projective base $\{a,b,c,d\}$ (giving a coordinate system) in the projective plane, cf. [a2], Sects. 16.5.5.1, 16.7.5.

References

[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a2] M. Berger, "Geometry" , 1–2 , Springer (1987) pp. Sects. 10.11.3, 17.5.4 (Translated from French)
[a3] O. Veblen, J.W. Young, "Projective geometry" , II , Blaisdell (1946) pp. 169; 233
How to Cite This Entry:
Nine-point circle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nine-point_circle&oldid=31775
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article