RO-NMO 2000 Problem 92

Let $A,B$ be two points in a plane and let two numbers $a,b\in (0,1) .$ For each point $M$ that is not on the line $AB$ consider $P$ on the segment $AM$ and $N$ on $BM$ (both excluding the extremities) such that $BN=b\cdot BM$ and $AP=a\cdot AM.$ Find the locus of the points $M$ for which $AN=BP.$