perplexus.info :: Geometry : SQUARE<sup>COMPASS</sup>

SQUARE^COMPASS (Posted on 2011-01-30)

For the following use a compass and straightedge procedure.

Given a unit square ABCD construct 4 circles centered at each corner and having diagonal radius. Form the smallest of the squares defined by the points where these circles intersect.

Use this process on the new square to create a third square.

1. Find and provide the area [in surd format] of each square so formed.

2. Following this process find a general expression for the area of the n^th square.

Submitted by brianjn

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Solution: (Hide)

Let me take you to Charlie's derivation, he has explained it far better than I could.

In comments I have noted that Charlie's general formula looks different to what I derived.

The area of my first constructed square was:
(4 - √7) and the second (4 - √7)².

The generalised format: (4 - √7)ⁿ takes all constructed squares into account but ignores the very first unit square.

Rewriting the expression as:

(4 - √7)^n-1

brings this expression into full agreement with the formula offered by Charlie:
(sqrt(7) - 1)^(2*(n-1)) / 2^(n-1),
or
(√7 -1)^2*(n-1) / 2^(n-1)

I wish to acknowledge Jer's valued assistance in helping consolidate the basic foundation for this problem.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: solution	brianjn	2011-01-31 19:20:14
re(2): solution	Charlie	2011-01-31 10:20:10
re: solution	brianjn	2011-01-31 05:51:50
solution	Charlie	2011-01-30 20:03:38

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