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Antoine gave a good and simple proof for his answer to Frank's puzzle.
Then he concluded with the challenge:

"Standing by for easier and more immediate explanations ... :-))"

Picking up the glove, I propose:

"The MPP (in Frank's setup) is exactly at half the distance of the third
LOP from the intersection of the orthogonal ones, because the Lemoine
point of an orthogonal triangle bisects the altitude to the hypotenuse."

(That the premise is correct you will immediately see when you draw an
orthogonal triangle, its altitude to the hypotenuse, and a median as
well as a symmedian from one of the acute corners. The altitude cuts the
original triangle into two similar ones. You will see that the symmedian
of the large triangle coincides with the median of the small one on the
side of the chosen corner.)

I am fully aware that my proposed "proof" depends on the assumption that
the reader already takes for granted that the MPP is found by
intersecting the symmedians. It is the sole purpose of this post to once
more draw attention to this little known fact.

To those who will find my "proof" neither easier nor more immediate than
Antoine's I will respond that what you find easy depends on what you
already know. For example, Antoine concludes his proof with the bold
assertion

"Since the sum " up + us " is constant (and equal to d), the quantity "
up**2 + us**2 " is minimum when up = us."

Well, who says? Most of us know that, but strictly, it should not be
taken for granted. This is exactly the place where the differential
calculus is hidden that Antoine wanted to avoid in the first place.

Not to speak of Frank's analysis that starts out thus:

"Any pair of LOPs with estimated errors on them yields an error ellipse. "

Wow!

If Antoine and Frank get away with this, it is because they are relying
on a certain level of common knowledge amongst their readers. I wish
that the true location of the MPP would equally become such common
knowledge. Then my "proof" would look only half as facetious as it
probably does now. More importantly, it would save us much speculation.

Herbert Prinz

   

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