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Topic: Three points on incircle
Replies: 5   Last Post: May 29, 2014 3:04 PM

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 Avni Pllana Posts: 546 Registered: 12/6/04
Re: Three points on incircle
Posted: May 29, 2014 3:04 PM

> > >
> > > The next problem is as follows:
> > >
> > > Let A=[-1,-1], B=[1,-1]. Construct points D, E, F

> > on
> > > the unit circle, such that line DE is parallel to
> > the
> > > y-axis, and triples {B,E,F], {A,F,D} are
> collinear
> > > respectively.
> > >

> > Hi Avni,
> >
> > Construct the unit circle and points A and B
> > Construct P(2/5,1/5) with OP=1/sqrt(5)
> > Circle centre O, radius OP meets ox in

> Q(1/sqrt(5),0)
> > Vertical thru Q meets unit circle at D and E,
> > with y = +- 2/sqrt(5)
> > Line AD meets unit circle at F(-sqrt(5)/3,-2/3)
> > B,E,F are collinear.
> >
> > Regards, Peter Scales.

>
> Hi Peter,
>
> excellent. The construction is incredibly simple, we
> have only to draw a line through the origin with the
> slope 2/1, an it intersects the unit circle at point
> D.
>
> The structure again has some interesting properties:
>
> 1) The golden ratio FD/FE =(sqrt(5)+1)/2 , and
> 2) angle(ABF)=angle(ODF).
>
> P.S.: This problem is a special case when vertex C of
> triangle ABC lies at infinity. Solving this problem
> for an arbitrary triangle ABC is left to the reader
> ;-)
>
>
> Best regards,
> Avni

The solution of the problem is shown in the following link

http://trisectlimacon.webs.com/incircle3p.pdf

Best regards,
Avni

Date Subject Author
4/27/14 Avni Pllana
4/28/14 Peter Scales
4/28/14 Avni Pllana
4/30/14 Peter Scales
4/30/14 Avni Pllana
5/29/14 Avni Pllana