
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 > > > yaxis, 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

