Find the mid point of a line
segment
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Let A , B be two given points. Use a pair of
compasses only (ruler and other instruments not allowed) to construct
the mid-point of the line segment A B. |
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Solution: For simplicity, let AB = a. Use A, B as centers, a as radii, draw two circles and let C be their
point of intersection. On circle center B
construct points D and E such that CD = DE = a. Then AE = 2a. Use E as center, EA as
radius, construct a circle which cuts circle center A at F and G. Use F and G as centers,
FA as radius, draw two arcs, which cut one another at M. M is the mid-point of
AB. |
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Reasoning From symmetry, M is on AB. It is easy to note that ÐFMA = ÐFAM = ÐEFA. and DAFM ~ DAEF (AAA) \ |
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Mathematicians Lorenzo
Mascheroni (1750 – 1800) and Georg
Mohr (1640 – 1697) independently proved that all Euclidean
constructions can be made with compasses alone, so a straight edge in not
needed. |
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