Nicomedes invented a curve named "conchoid" to solve the Delian problem.
It is used to find a point K on X-axis such that line FK passes through
a point "H" on line CZ, and cuts x-axis while HK = CF is being satisfied.
So this process looks similar to "Verging"(neusis) used
by Archimedes, Pappus, and many Greek mathematicians in angle trisection.
Make rectangle ABCL. D,E are mid-points of AB and BC, Extend LD to produce G on x-axis. Erect EF perpendicular to x-axis and make CF = AD. Draw CZ parallel to GF. Choose a point K on x-axs such that HK = CF. Such a point is the intersection of special curve "conchoid" and x-axis. This curve can be drawn when F is "pole", CZ is "rule" and CF is "distance". Alternative is use a ruler marked with length CF on the edge, and find the location of FK (verging,or neusis process) Once point K is found, extend line KL to produce point M on y-axis. Then AM = {AB.BC^{2}}^{1/3} and CK = {AB^{2}.BC}^{1/3} |
You can see the process using "conchoid" in animation.
******** Nicomedes_Delian_desc.dwg *******
To create this drawing and animation:
Load Nicomedes_Delian.lsp (load "Nicomedes_Delian")
Then from command line, type test_0
To create this drawing and animation:
Load Nicomedes_Delian.lsp (load "Nicomedes_Delian")
Then from command line, type Nicomedes_Delian
Heath gives the following proof: First step is to show MD = FK. Since E bisects BC, BK.KC + CE^{2} = KF^{2} (note: recall (a + b) (a - b) = a^{2} + b^{2} ) Add EF^{2} to both sides; BK.KC + CF^{2} = KF^{2} (1) Since AL is parallel to GK, so is CL to MB, MA / AB = ML / LK = BC / CK But AB = 2 AD, and BC = (1/2)GC Therefore MA / AD = GC /CK = FH / HK and MD / DA = FK / HK Here,DA = HK,therefore MD = FK Since D bisects AB, MD^{2} = BM.MA + DA^{2} Combining this with (1) because MD = FK, BM.MA + DA^{2} = BK.KC + CF^{2} But DA = CF; therefore BM.MA = BK.KC And CK/MA = BM/BK = LC/CK, Whereas, in triangles MAl & MBK, BM/BK = MA/AL Therefore LC/CK = CK/MA = MA/AL or AB/CK = CK/MA = MA/BC Then AM = {AB.BC^{2}}^{1/3} and CK = {AB^{2}.BC}^{1/3} |
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Last Updated Nov 22, 2006
Copyright 2006 Takaya Iwamoto All rights reserved. .