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On 2010-12-10 02:44, John Karl wrote:
>
> Frank & Herbert,
>
> I can see in the Villarceau article (without any knowledge of French)
> that he obtains the construction of the symmedian point that Prince
> mentions. But does he show that it minimizes the sum of the squared
> distances to the three sides?
>
Karl,

As I said before:

"When solving the minimum condition by partial differentiation,
Villarceau found the solution to be at the point for which the distances
to the sides of the triangle are as the sides of the triangle
themselves. (In other words x:y:z = a:b:c)"

So, yes, he did.

And I further said:

"This is a defining property of the symmedian point. Lemoine showed
independently, but around the same time, that the symmedian point has
the property of minimizing the square sum of the distances."

So, Lemoine did, too.

Herbert Prinz

   

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