Re: [EMHL] A special line in quadrilateral
- Dear friends, Nikolaos Dergiades proposed the following corrected construction. This construction completely resolves the problem.
Construction by Nikolaos Dergiades:
"So if ABCD is a convex quadrilateral
with diagonal intersection O and
OA = a, OB = b, OC = c, OD = d
you want to construct a line AK (K on AB)
such that f = sin(KOB)/sin(KOA)
and f^2 = (1/b^2 + 1/d^2)/(1/a^2 + 1/c^2).
So my construction is modified as
Let D' be the rotation of D about O by angle 90.
Let P be the orthogonal projection of O on BD'.
Let B' be a point on the cemiline OB such that OB' = OP
Let C' be the rotation of C about O by angle 90.
Let Q be the orthogonal projection of O on AC'.
Let A' be a point on the cemiline OA such that OA' = OQ.
If S is the mid-point of A'B' then the line OS meets
AB at the point K."
Thank you Nikolaos!
--- In Hyacinthos@yahoogroups.com, Nikolaos Dergiades <ndergiades@...> wrote:
> Dear Yakub,
> I think that in your study you are
> interested for f = sin(KOA)/sin(KOB)
> and not for f = sin(KOB)/sin(KOA).
> I don't have an explicit geometric meaning
> but a geometric construction.
> Let D' be the rotation of D about O by angle 90.
> Let P be the orthogonal projection of O on BD'.
> Let A' be a point on the cemiline OA such that OA' = OP
> Let C' be the rotation of C about O by angle 90.
> Let Q be the orthogonal projection of O on AC'.
> Let B' be a point on the cemiline OB such that OB' = OQ.
> If S is the mid-point of A'B' then the line OS meets
> AB at the point K.
> Best regards
> Nikos Dergiades
> > Dear friends, I ask you to find
> > explicit geometric meaning for the following algebraically
> > defined construction of special line through intersection
> > point O of diagonals AC and BD of convex quadrilateral ABCD.
> > Let K and M be points on sides AB and CD, respectively, so
> > that KM passes through O. Suppose that
> > f=sin(angle(KOB))/sin(angle(KOA)). Let
> > f^2 = (1/BO^2+1/DO^2)/(1/AO^2+1/CO^2). Are there any
> > explicit geometric construction for this line KM? This line
> > appeared in my study of some inequality in quadrilateral.
> > See [1, section 3].
> >  Y. N. Aliyev, New inequalities on triangle areas,
> > Journal of Qafqaz University, Number 25, 2009,
> > 129-135. (See the link:
> > http://www.qafqaz.edu.az/journal/20092516nev.pdf )
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