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Ceva's and Menelaus's theorems

Properties of figures

Japanese school year: Math A

What you learn

Learn ratio relationships that hold when three lines through triangle vertices meet at one point, or when a line intersects three sides or their extensions. Using similarity and ratios, these theorems efficiently compute complex line segments. They are widely utilized in analyzing geometric figures.

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Key points

Known as Ceva's theorem, this relates side ratios when three lines drawn from vertices meet at a single point. Multiplying the divided ratios around points P,Q,RP, Q, R always equals 1.

BPPC⋅CQQA⋅ARRB=1\frac{BP}{PC} \cdot \frac{CQ}{QA} \cdot \frac{AR}{RB} = 1

Known as Menelaus' theorem, this relates segment ratios when a line intersects triangle sides (or extensions) at D,E,FD, E, F. Multiplying the ratios sequentially around the triangle yields 1.

AFFB⋅BDDC⋅CEEA=1\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1

This method prevents confusing numerators and denominators in Ceva's or Menelaus' formulas. Follow the perimeter in a loop: "vertex →\to division point →\to vertex", writing segments alternately.

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