Re: [HM] Malfatti's problem


Subject: Re: [HM] Malfatti's problem
From: Antreas P. Hatzipolakis (xpolakis@otenet.gr)
Date: Thu Mar 23 2000 - 03:58:08 EST


[Kotera Hiroshi]
>
> In 1803, G.F.Malfatti proposed the problem of cutting three right
> circular cylinders and of maximum total volume from a given right
> triangular prism.
>

[John Conway]
>
> Let's call this problem (1).
>

[Kotera Hiroshi]
>
> The problem was intuitively reduced to the following:
> To inscribe three circles in a given triangle so that each circle
> will be tangent to two sides of the triangle and to the other two
> circles.
>

[John Conway]
>
> Let's call this problem (2).
>

[Kotera Hiroshi]
>
> In "A Survey of Geometry", H.Eves said that Malfatti gave a prolix
> and incomplete analytical solution of the reduced problem.
>

[Antreas P. Hatzipolakis]
>
> I don't know which is H. Eves' source, but mine doesn't say that
> Malfatti's solution was wrong.
>

[John Conway]
>
> What Malfatti did correctly was to solve problem (2). But this
> doesn't happen to give the solution to problem (1), for which
> the incircle is one of the 3 circles involved. Get it??
>

[Antreas P. Hatzipolakis]

.................................................. Sure!! :-)

The firsts, who pointed out that problem (1) is not equivalent to problem (2),
were Lob and Richmond [1], by the counterexample of the equilateral triangle.
Michael Goldberg [2,3] showed that the solution of Problem (2) is NEVER the
best solution for the Problem (1). As Ogilvy [4, p. 147] remarks:
"Goldberg's conclusions are based on calculations and graphs; a purely
mathematical proof is doubtless difficult and has not yet been published."
It was not until 1992 that a COMPLETE solution of (1) appeared by Zalgaller
and Los' [5].

References:

[1] Lob, H. - Richmond, H. W.: On the Solution of Malfatti's Problem for
a Triangle.
Proc. London Math. Soc. 2(1930) 287-304.

[2] Goldberg, M.: On the Original Malfatti Problem.
Math. Mag. 40(1967) 241-247.

[3] Goldberg, M.: The Converse Malfatti Problem.
Math. Mag. 41(1968) 262-266.

[4] Ogilvy, C. Stanley: Excursions in Geometry. Dover, 1990
[First publ.: Oxford U. Press, 1969]

[5] Zalgaller, V.A. - Los', G.A.: Solution of the Malfatti Problem. (Russian)
Ukr. Geom. Sb. 35, 14-33 (1992).
Translation: J. Math. Sci., New York 72, No.4, 3163-3177 (1994)
The authors give the solution of the Malfatti problem to place three non-
overlapping discs with maximal summary of areas into a triangle. \par The
main result: Let $2\alpha$, $2\beta$, $2\gamma$ be angles of the triangle
$ABC$, where $0 < \alpha \leq \beta \leq \gamma \geq {\pi \over 2}$. Let
$K\sb 1$ be a circle inscribed into a triangle $ABC$, $K\sb 2$ be a circle
tangent to $AB$, $AC$ and $K\sb 1$, $K\sb 3$ be a circle either tangent to
$AB$, $BC$ and $K\sb 1$, for $\sin \alpha = tg {\beta \over 2}$, or tangent
to $AB$, $AC$ and $K\sb 2$. The discs bounded by circles $K\sb 1$, $K\sb
2$, $K\sb 3$ are solution of the Malfatti problem.
[ P.Burda (Ostrava) ]
(From Zbl)

Antreas



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