Mass points
Mass points assign a “weight” to each vertex of a triangle so that a cevian's foot balances like a lever: if a segment from to a point on divides it so that , assign mass to and mass to — then 's mass is the sum, and it “balances” along . Chaining this across multiple cevians turns a ratio-finding problem into simple weighted averaging, with no coordinates required.
Ceva's theorem
Three cevians (one from each vertex, to a point on the opposite side) are concurrent (all pass through one point) if and only if:
Menelaus's theorem
Menelaus's theorem uses the exact same product, but for a very different setup: a single transversal line crossing the (possibly extended) three sides of a triangle at points . Those three points are collinear if and only if:
The two theorems are easy to mix up because the formula is identical — the difference is entirely in what's being tested: three concurrent cevians (Ceva) versus three collinear pointson the extended sides (Menelaus).
Worked example
Problem: In , cevians are concurrent, with and . Find.
Solution: By Ceva's theorem:
Practice problems
1.Cevians are concurrent with , . Find .
2.A median, by definition, has its foot at the midpoint of a side. Explain why the three medians automatically satisfy Ceva's theorem.
3.A transversal crosses the sides (or extensions) of a triangle with and . Find for the points to be collinear (Menelaus).
4.What is the key difference between when you'd apply Ceva's theorem versus Menelaus's theorem?