In triangle ABC the point D lies on BC with BD : DC = 3 : 2, and the point E lies on CA with CE : EA = 4 : 3. The point F lies on AB, and the three cevians AD, BE and CF all pass through one common point. What is the ratio AF : FB?

Ceva's theorem, with the three ratios taken in the same rotational order round the triangle, says (BD/DC) × (CE/EA) × (AF/FB) = 1. Substituting, (3/2) × (4/3) × (AF/FB) = 1, and since (3/2) × (4/3) = 2 this leaves AF/FB = 1/2, so AF : FB = 1 : 2. The trap 2 : 1 writes the last ratio upside down — the numerator must be the piece next to A, matching the pattern set by BD (next to B) and CE (next to C). Multiplying the ratios in the wrong order gives 3 : 4, and 1 : 4 comes from halving twice.