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 Geometric calculator (Posted on 2006-11-24)
Given three segments of length 1, a and b in the plane, how can one construct segments of length a+b, |a-b|, ab, a/b, √a using ruler and compass? Which other calculator functions can be performed by geometric construction?

 See The Solution Submitted by JLo Rating: 4.0000 (1 votes)

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 Solution | Comment 1 of 3
`Construct concentric circles (radii a and b) with aline through their center intersecting the circles atpoints A, B, C, and D in that order. Then,`
`  |AC| = |BD| = a+b   and   |AB| = |CD| = |a-b|.`
`Construct rays ABD and ACE separated by an angle ofapproximately 60 degrees such that DE is parallel to BC.Triangles ABC and ADE are similar. Therefore,`
`   |AB|     |AD|     |AB| + |BD|  ------ = ------ = -------------   |AC|     |AE|     |AC| + |CE|`
`Thus,`
`          |AC||BD|  |CE| = ----------            |AB|`
`Cases:`
`  |AC|  |BD|  |AB|       |CE| -----------------------------    a     b     1         ab    a     1     b         a/b    1     1     a         1/a`
`Construct a circle with diameter |AB| such that|AB| = a+1. Construct a point C on diameter ABsuch that |AC| = a. Construct a line through pointC perpendicular to AB intersecting the circle atpoints D and E. Then,`
`  |CD| = |CE| = sqrt(a).`
` `

 Posted by Bractals on 2006-11-24 10:42:24
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