
Re: Three points on incircle
Posted:
Apr 30, 2014 3:50 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

