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 .

HyperbolaProperty.html

HyperbolaWRAsymptotics.html

Produced with EucliDraw© |