**Subject: **Re: [HM] Malfatti's problem

**From: **Antreas P. Hatzipolakis (*xpolakis@otenet.gr*)

**Date: **Thu Mar 23 2000 - 03:58:08 EST

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[Kotera Hiroshi]

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

*>
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*> Let's call this problem (1).
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*>
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[Kotera Hiroshi]

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

*>
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*> Let's call this problem (2).
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*>
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[Kotera Hiroshi]

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

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

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