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 2015-07-20 10:04:08