Yesterday browsing in this marvelous site accidentally I ended on this page and took a look on the proof posted, I always learn new or something interesting. I challenge myself to find other proof more synthetic, below is my proposed proof based on projective geometry without drawing a line or writing an equation just reaoning, I appreciate take a look and tell me if it is correct of it has a flaw.

As the ratio r express the absolute value of ratio of the segments AP/BP, there are two point M and N on the straight line AB which satisfy the above condition. Indeed r and minus r are the barycentric coordinates of M and N with respect to to A and B on line AB or in other words the simple ratio ABM and ABN are equal an opposed in sign.
Therefore the cross ratio is harmonic being M and N conjugated.

So if we draw a pencil of rays from any arbitrary point in the plane to the points A,B,M, and N the harmonic ratio will be preserved, however since point P must be selected so ray PM be the bisector of angle APB, the ray PN must be the other exterior bisector of the angle APB. All the above means that the angle NPM must be always 90 degrees and since MN is constant P must belong to a circumference of diameter MN.