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Owner confidence rating: High Entry average rating: No information on entry rating
isogonal conjugate (Definition)

Let $\triangle ABC$ be a triangle, $AL$ the angle bisector of $\angle BAC$ and $AX$ any line passing through $A$. The isogonal conjugate line to $AX$ is the line $AY$ obtained by reflecting the line $AX$ on the angle bisector $AL$.

\includegraphics{isogonal.eps}
In the picture $\angle YAL = \angle LAX$. This is the reason why $AX$ and $AY$ are called isogonal conjugates, since they form the same angle with $AL$. (iso= equal, gonal = angle).

Let $P$ be a point on the plane. The lines $AP,BP,CP$ are concurrent by construction. Consider now their isogonals conjugates (reflections on the inner angle bisectors). The isogonals conjugates will also concurr by the fundamental theorem on isogonal lines, and their intersection point $Q$ is called the isogonal conjugate of $P$.

If $Q$ is the isogonal conjugate of $P$, then $P$ is the isogonal conjugate of $Q$ so both are often referred as an isogonal conjugate pair.

An example of isogonal conjugate pair is found by looking at the centroid of the triangle and the Lemoine point.



"isogonal conjugate" is owned by drini. [ owner history (1) ]
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See Also: symmedian, Lemoine point, fundamental theorem on isogonal lines

Also defines:  isogonal conjugate pair, isogonal
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Cross-references: Lemoine point, centroid, intersection, fundamental theorem on isogonal lines, reflections, concurrent, plane, point, angle, passing through, line, angle bisector, triangle
There are 3 references to this entry.

This is version 4 of isogonal conjugate, born on 2002-09-02, modified 2003-08-19.
Object id is 3406, canonical name is IsogonalConjugate.
Accessed 4652 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
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Fields of characteristic p, with p a prime by mbhatia on 2006-12-02 06:49:59
Let F be a field of characteristic p not equal to 0. Let g be a function from F to F defined by g(a)=a^p for all a in F.

It is easy to show that g is a one-to-one ring homomorphism of F into itself.

Is there an example of a field F where g (defined above) is NOT onto?
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