Let three similar Isosceles Triangles
,
, and
be constructed on the sides of a Triangle
. Then the Envelope of the axis of the Triangles
and
is Kiepert's parabola, given by
![\begin{displaymath}
{\sin A(\sin^2 B-\sin^2 C)\over u}+{\sin B(\sin^2 C-\sin^2 A)\over v}+{\sin C(\sin^2 A-\sin^2 B)\over w}=0
\end{displaymath}](k_506.gif) |
(1) |
![\begin{displaymath}
{a(b^2-c^2)\over u}+{b(c^2-a^2)\over v}+{c(a^2-b^2)\over w}=0,
\end{displaymath}](k_507.gif) |
(2) |
where
are the Trilinear Coordinates for a line tangent to the parabola. It is tangent to the sides of
the Triangle, the line at infinity, and the Lemoine Line. The Focus has Triangle Center
Function
![\begin{displaymath}
\alpha=\csc(B-C).
\end{displaymath}](k_509.gif) |
(3) |
The Euler Line of a triangle is the Directrix of Kiepert's parabola. In fact, the
Directrices of all parabolas inscribed in a Triangle pass through the
Orthocenter. The Brianchon Point for Kiepert's parabola is the
Steiner Point.
See also Brianchon Point, Envelope, Euler Line, Isosceles Triangle,
Lemoine Line, Steiner Points
© 1996-9 Eric W. Weisstein
1999-05-26