KURSCHAK 2021 Problem 1

Let $P_0=(a_0,b_0),P_1=(a_1,b_1),P_2=(a_2,b_2)$ be points on the plane such that $P_0P_1P_2\Delta$ contains the origin $O$ . Show that the areas of triangles $P_0OP_1,P_0OP_2,P_1OP_2$ form a geometric sequence in that order if and only if there exists a real number $x$ , such that $$a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0$$