lunes, 21 de julio de 2014

A metric relationship between Fermat points and isodynamic points

Let $A_1BC$, $AB_1C$, $ABC_1$ be the equilateral triangles erected outwards the triangle $ABC$ and $A_2BC$, $AB_2C$, $ABC_2$  the equilateral triangles erected inwards the triangle $ABC$. The Fermat points $X_{13}$ and $X_{14}$ are the perspectors of $ABC$ and the triangles $A_1B_1C_1$ and $A_2B_2C_2$ respetively. We consider the equal distances  $d_{13}=AA=BB_1=CC_1$ and $d_{14}=AA_2=BB_2=CC_2$.

On the other hand, we consider the points $X_{15}$ and $X_{16}$, that are the isogonal conjugates of Fermat points and also are the only points that have equilateral pedal triangles. If we call $l_{15}$ and $l_{16}$ the sidelengths of the corresponding pedal triangles, then we have the formula:

$d_{13} l_{15} = 2 \Delta = d_{14} l_{16}$,

where $\Delta$ is the area of the triangle $ABC$.

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