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Part of 2021 Kurschak Competition

Problems(1)

Areas forming geometric sequence

Source: Kürschák József Mathematical Competition 2021/1

10/8/2021
Let P0=(a0,b0),P1=(a1,b1),P2=(a2,b2)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 P0P1P2ΔP_0P_1P_2\Delta contains the origin OO. Show that the areas of triangles P0OP1,P0OP2,P1OP2P_0OP_1,P_0OP_2,P_1OP_2 form a geometric sequence in that order if and only if there exists a real number xx, such that a0x2+a1x+a2=b0x2+b1x+b2=0 a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0
geometryalgebrageometric sequence