Triangle PQR has incentre I and circumcentre O. Suppose that ∠ PIO = 90^{o} and ∠ QIO = 45^{o}.
Find the ratio PQ: QR : RP
(In reply to
Solution by Bractals)
I concur with your solution but you don't seem to indicate the 5:4:3 is the only ratio that works. But from my sketch it is.
Some interesting things happened when I drew this on geometer's sketchpad:
I coordinatized and made I=(0,0), O=(1,0), P=(0,p), Q=(q,q)
Constructing the circumcenter fixes Q (q is some function of p)
With line PQ known you can construct the incenter and the third corner of a triangle that has this incenter. The point will only be on the circumcenter if P is in just the right spot.
It turns out (unproven) that P=(0,2), Q=(2,2), R=(6/5,2/5)
PR = 6√5/5
RQ = 8√5/5
PQ = 10√5/5
So the ratio is 5:4:3

Posted by Jer
on 20150720 10:04:08 