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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 $A _ { 1 } A _ { 2 } A _ { 3 }$ and a line $L_1$ from one of the vertices, say from $A _ { 1 }$, to the opposite side. The corresponding isogonal line $L _ { 1 } ^ { \prime }$ is obtained by reflecting $L_1$ with respect to the [[Bisectrix|bisectrix]] in $A _ { 1 }$.

If the lines $L _ { 1 } = A _ { 1 } P _ { 1 }$, $L _ { 2 } = A _ { 2 } P _ { 2 }$ and $L _ { 3 } = A _ { 3 } P _ { 3 }$ are concurrent (i.e. pass through a single point $X$, i.e. are [[Cevian lines]]), then so are the isogonal lines $L _ { 1 } ^ { \prime }$, $L _ { 2 } ^ { \prime }$, $L _ { 3 } ^ { \prime }$. This follows fairly directly from the [[Ceva theorem|Ceva theorem]]. The point $X ^ { \prime } = L _ { 1 } ^ { \prime } \cap L _ { 2 } ^ { \prime } = L _ { 2 } ^ { \prime } \cap L _ { 3 } ^ { \prime } = L _ { 1 } ^ { \prime } \cap L _ { 3 } ^ { \prime }$ is called the isogonal conjugate point. If the [[Barycentric coordinates|barycentric coordinates]] of $X$ (often called trilinear coordinates in this setting) are $( \alpha : \beta : \gamma )$, then those of $X ^ { \prime }$ are $( \alpha ^ { - 1 } : \beta ^ { - 1 } : \gamma ^ { - 1 } )$

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

Figure: i130080a

Another notion in rather the same spirit is that of the isotomic line to $L_1$, which is the line $L_1 ^ { \prime \prime } = A _ { 2 } P ^ { \prime \prime }_1$ such that $| A _ { 2 } P _ { 1 } ^ { \prime \prime } | = | P _ { 1 } A _ { 3 } |$. Again it is true that if $L_1$, $L_{2}$, $L_3$ are concurrent, then so are $L _ { 1 } ^ { \prime \prime }$, $L _ { 2 } ^ { \prime \prime }$, $L _ { 3 } ^ { \prime \prime }$. This follows directly from the [[Ceva theorem|Ceva theorem]].

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

Figure: i130080b

The point $X ^ { \prime \prime } = L _ { 1 } ^ { \prime \prime } \cap L _ { 2 } ^ { \prime \prime } = L _ { 2 } ^ { \prime \prime } \cap L _ { 3 } ^ { \prime \prime } = L _ { 1 } ^ { \prime \prime } \cap L _ { 3 } ^ { \prime \prime }$ is called the isotomic conjugate point. The barycentric coordinates of $X ^ { \prime \prime }$ are $( a ^ { 2 } \alpha ^ { - 1 } : b ^ { 2 } \beta ^ { - 1 } : c ^ { 2 } \gamma ^ { - 1 } )$, where $a$, $b$, $c$ 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 $X \mapsto X ^ { \prime }$ and $X \mapsto X ^ { \prime \prime }$, i.e. isogonal conjugation and isotomic conjugation, are better regarded as involutions of the projective plane $\mathbf{P} ^ { 2 } ( \mathbf{R} )$, [[#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>