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The compass equivalence theorem is an important statement in compass and straightedge constructions. In these constructions it is assumed that whenever a compass is lifted from a page, it collapses, so that it may not be directly used to transfer distances. While this might seem a difficult obstacle to surmount, the compass equivalence theorem states that any construction via a "fixed" compass may be attained with a collapsing compass. In other words, it is possible to construct a circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements. ==Construction== We are given points A, B, and C, and wish to construct a circle centered at A with the same radius as BC (the first green circle). *Draw a circle centered at A and passing through B and vice versa (the red circles). They will intersect at point D and form equilateral triangle ABD. *Extend DB past B and find the intersection of DB and the circle BC, labeled E. *Create a Circle centered at D and passing through E (the blue circle). *Extend DA past A and find the intersection of DA and the circle DE, labeled F. *Construct a circle centered at A and passing through F (the second green circle) *Because ADB is an equilateral triangle, DA = DB. *Because E and F are on a circle around D, DE = DF. *Therefore, AF = BE. *Because E is on the circle BC, BE = BC. *Therefore, AF = BC. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Compass equivalence theorem」の詳細全文を読む スポンサード リンク
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