Maximal Properties of the Pythagorean Relation

A 1893 sangaku from the Fukusima prefecture [Fukagawa, Pedoe, problem 2.3.6] admits a relatively straightforward computational solution, but also an elegant solution that practically avoids any kinds of computations.

Rectangles cut off from a right triangle leave three triangles with inradii r1, r2, r3, in the increasing order. Show that when the area of the rectangle is a maximum possible then

(r1)² + (r2)² = (r3)².

Solution

References

  1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

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Rectangles cut off from a right triangle leave three triangles with inradii r1, r2, r3, in the increasing order. Show that when the area of the rectangle is a maximum possible then

(r1)² + (r2)² = (r3)².

Solution

The four triangles ABC, EDC, AEF, DBG are obviously similar. Their inradii relate exactly like any of the corresponding sides. For the definiteness sake, let's focus on the hypotenuse.

If the sides of ΔABC are a = BC, b = AC, c = AB, then, for some k > 0, the corresponding sides in smaller triangles are:

  ABC a b c
  EDC ka kb kc
  AEF ... ... (1-k)b
  DBG ... ... (1-k)a

We want to show that, for k, for which the area of the rectangle DEFG is a maximum,

[(1 - k)a]² + [(1 - k)b]² = (kc)²,

or

(1) (1 - k)²·a² + (1 - k)²·b² = k²·c².

However, we also have

a² + b² = c².

Multiplying (2) by k² and subtracting from (1) we get

(1 - 2k)·(a² + b²) = 0

which is only possible when k = 1/2. This is then bound to be the optimal configuration implied by the sangaku. To proceed, we need to fill some of the empty cells in the table above:

DG / AC = BD / AB,

so that DG = (1 - k)ab/c. Since DE = kc, the area S of the rectangle DEFG is given by

S = DG × DE = (1 - k)k·ab

which attains its maximum along with f(k) = (1 - k)k. This is an inverted parabola with the roots at 0 and 1, and the maximum the midway between the two, i.e., at k = 1/2.

Sangaku

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