View source for Isogonal

Literally  "same angle" . There are several concepts in mathematics involving isogonality.

===Isogonal trajectory.===
A trajectory that meets a given family of curves at a constant angle. See [[Isogonal trajectory|Isogonal trajectory]].

===Isogonal mapping.===
A (differentiable) mapping that preserves angles. For instance, the [[Stereographic projection|stereographic projection]] of cartography has this property [[#References|[a2]]]. See also [[Conformal mapping|Conformal mapping]]; [[Anti-conformal mapping|Anti-conformal mapping]].

===Isogonal circles.===
A circle is said to be isogonal with respect to two other circles if it makes the same angle with these two, [[#References|[a1]]].

===Isogonal line.===
Given a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300801.png" /> and a line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300802.png" /> from one of the vertices, say from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300803.png" />, to the opposite side. The corresponding isogonal line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300804.png" /> is obtained by reflecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300805.png" /> with respect to the [[Bisectrix|bisectrix]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300806.png" />.

If the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i1300809.png" /> are concurrent (i.e. pass through a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008010.png" />, i.e. are [[Cevian lines]]), then so are the isogonal lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008013.png" />. This follows fairly directly from the [[Ceva theorem|Ceva theorem]]. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008014.png" /> is called the isogonal conjugate point. If the [[Barycentric coordinates|barycentric coordinates]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008015.png" /> (often called trilinear coordinates in this setting) are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008016.png" />, then those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008017.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008018.png" />

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i130080a.gif" />

Figure: i130080a

Another notion in rather the same spirit is that of the isotomic line to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008019.png" />, which is the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008021.png" />. Again it is true that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008024.png" /> are concurrent, then so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008027.png" />. This follows directly from the [[Ceva theorem|Ceva theorem]].

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i130080b.gif" />

Figure: i130080b

The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008028.png" /> is called the isotomic conjugate point. The barycentric coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008029.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008033.png" /> are the lengths of the sides of the triangle. The [[Gergonne point|Gergonne point]] is the isotomic conjugate of the [[Nagel point|Nagel point]].

The involutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008035.png" />, i.e. isogonal conjugation and isotomic conjugation, are better regarded as involutions of the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130080/i13008036.png" />, [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''I''' , Springer  (1987)  pp. 327</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Hilbert,   S. Cohn-Vossen,   "Geometry and the imagination" , Chelsea  (1952)  pp. 249</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.A. Johnson,   "Modern geometry" , Houghton–Mifflin  (1929)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.H. Eddy,   J.B. Wilker,   "Plane mappings of isogonal-isotomic type"  ''Soochow J. Math.'' , '''18''' :  2  (1992)  pp. 135–158</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Altshiller–Court,   "College geometry" , Barnes &amp; Noble  (1952)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H.S.M. Coxeter,   "The real projective plane" , Springer  (1993)  pp. 197–199  (Edition: Third)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F. Bachmann,   "Aufbau der Geometrie aus dem Spiegelungsbegriff" , Springer  (1973)  (Edition: Second)</TD></TR></table>