Compass equivalence theorem

<h2 id="construction">Construction</h2>

The following construction and proof of correctness are given by Euclid in his Elements.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Although there appear to be several cases in Euclid's treatment, depending upon choices made when interpreting ambiguous instructions, they all lead to the same conclusion,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> and so, specific choices are given below.
Given points A, B, and C, construct a circle centered at A with radius the length of BC (that is, equivalent to the solid green circle, but centered at A).

<ul><li>Draw a circle centered at A and passing through B and vice versa (the red circles). They will intersect at point D and form the <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a> △ABD.</li>
<li>Extend DB past B and find the intersection of DB and the circle ◯BC, labeled E.</li>
<li>Create a circle centered at D and passing through E (the blue circle).</li>
<li>Extend DA past A and find the intersection of DA and the circle ◯DE, labeled F.</li>
<li>Construct a circle centered at A and passing through F (the dotted green circle)</li>
<li>Because △ADB is an equilateral triangle, DA = DB.</li>
<li>Because E and F are on a circle around D, DE = DF.</li>
<li>Therefore, AF = BE.</li>
<li>Because E is on the circle ◯BC, BE = BC.</li>
<li>Therefore, AF = BC.</li></ul>
<h2 id="alternative-construction-without-straightedge">Alternative construction without straightedge</h2>
It is possible to prove compass equivalence without the use of the straightedge. This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) in proofs of the <a href="/facts/Mohr%E2%80%93Mascheroni_theorem/N068VdqK">Mohr–Mascheroni theorem</a>, which states that any construction possible with straightedge and compass can be accomplished with compass alone.

Given points A, B, and C, construct a circle centered at A with the radius BC, using only a collapsing compass and no straightedge.

<ul><li>Draw a circle centered at A and passing through B and vice versa (the blue circles). They will intersect at points D and D'.</li>
<li>Draw circles through C with centers at D and D' (the red circles). Label their other intersection E.</li>
<li>Draw a circle (the green circle) with center A passing through E. This is the required circle.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
There are several proofs of the correctness of this construction and it is often left as an exercise for the reader.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Here is a modern one using <a href="/facts/Geometric_transformation/qRjrasJf">transformations</a>.

<ul><li>The line DD' is the <a href="/facts/Perpendicular_bisector/JTsya7Ca">perpendicular bisector</a> of AB. Thus A is the <a href="/facts/Reflection_(geometry)/5JHWnrql">reflection</a> of B through line DD'.</li>
<li>By construction, E is the reflection of C through line DD'.</li>
<li>Since reflection is an <a href="/facts/Isometry/K5wX8ah6">isometry</a>, it follows that AE = BC as desired.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Toussaint, Godfried T. (January 1993). "A New Look at Euclid's Second Proposition" (PDF). The Mathematical Intelligencer. 15 (3). Springer US: 12–24. doi:10.1007/bf03024252. eISSN 1866-7414. ISSN 0343-6993. S2CID 26811463. <a href="/wiki/Godfried_Toussaint" target="_blank">/wiki/Godfried_Toussaint</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Heath, Thomas L. (1956) [1925]. The Thirteen Books of Euclid's Elements (2nd ed.). New York: Dover Publications. p. 244. ISBN 0-486-60088-2. <a href="0-486-60088-2" target="_blank">0-486-60088-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Toussaint, Godfried T. (January 1993). "A New Look at Euclid's Second Proposition" (PDF). The Mathematical Intelligencer. 15 (3). Springer US: 12–24. doi:10.1007/bf03024252. eISSN 1866-7414. ISSN 0343-6993. S2CID 26811463. <a href="/wiki/Godfried_Toussaint" target="_blank">/wiki/Godfried_Toussaint</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Eves, Howard (1963), A survey of Geometry (Vol. I), Allyn Bacon, p. 185 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 212, ISBN 0-534-35188-3 <a href="0-534-35188-3" target="_blank">0-534-35188-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Eves, Howard (1963), A survey of Geometry (Vol. I), Allyn Bacon, p. 185 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 212, ISBN 0-534-35188-3 <a href="0-534-35188-3" target="_blank">0-534-35188-3</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Compass equivalence theorem open-in-new

Compass equivalence theorem