[alogo] Hyperbola Construction

Given two concurent lines DB, DC and a fixed point A construct all the parallelograms DEMI with two sides on these lines and a diagonal passing through A. Show that the locus of the opposite to A vertex M is a hyperbola and determine this in terms of the data of the lines DB, DC and point A.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

The following equation is a proposition in Euclid's elements (Book I, prop. 43): area(AJML) = area(IME)-area(ILA)-area(AJE) = area(IDE)-area(IFA)-area(AEG) = area(AFDG). Hence the product LM*MJ = area(AFDG)/sin(w) = DE*DG. Setting this equal to c^2/4 one can determine the hyperbola with respect to its asymptotics AF, AG (see HyperbolaWRAsymptotics.html ), characterized by the equation ML*MJ = c^2/4.
The figure has a relation to the one concering the minimization of the area of the triangle DEI, handled in the file EuclidMinimization.html . The triangle minimizing this area has points F, G and A as middles of its sides.
The hyperbola here is the same with the locus defined as follows: consider all secants IE through A of the sides of the fixed angle FDG and take M to be the symmetric w.r to the middle of IE. This is handled in the file HyperbolaProperty.html .

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