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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: 513
Registered: 12/6/04
Re: Three points on incircle
Posted: Apr 30, 2014 3:50 PM
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> >
> > 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



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