Cevians & Mass Points

Study Sheet

Cevians & Mass Points

Ratio-chasing: Ceva, Menelaus, masses, and areas

The Ratio-Chasing Recipes

Tip
Which theorem for which picture

Three cevians through ONE point: Ceva, BDDC⋅CEEA⋅AFFB=1\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}=1. One LINE crossing all three sides (one externally): Menelaus, same product (unsigned) equals 11.

In plain terms. Ceva is for concurrency, Menelaus is for a transversal line. In both, two known side-ratios always determine the third — draw the picture first and ask "one point or one line?"

Example. Cevians concurrent with BD:DC=4:5BD:DC=4:5 and AF:FB=2:3AF:FB=2:3: Ceva forces CE:EA=158CE:EA = \tfrac{15}{8}.

Tip
Mass points for ratios ALONG a cevian

Assign masses inversely to the side ratios so each cevian foot balances; then APPD=mass(D)mass(A)\dfrac{AP}{PD}=\dfrac{\text{mass}(D)}{\text{mass}(A)} at the intersection PP.

In plain terms. Put weights on the corners so every marked point is a balance point; the crossing point ratio reads off as mass-over-mass. Rescale one pair when two cevians disagree about a shared vertex.

Example. BD:DC=2:3BD:DC=2:3 and AF:FB=3:4AF:FB=3:4 give masses 4,3,24,3,2 at A,B,CA,B,C, so DD carries 55 and AP:PD=5:4AP:PD = 5:4.

Concept
Area ratios: same height, same base

Triangles sharing an apex have areas in the ratio of their bases; triangles on the same base have areas in the ratio of their heights. Chain these to convert cevian ratios into area ratios.

In plain terms. A cevian splits a triangle into two pieces whose areas are exactly the base ratio — walking around a figure multiplying such ratios computes almost any area fraction with no lengths at all.

Example. If BD:DC=2:3BD:DC = 2:3, cevian ADAD splits △ABC\triangle ABC into areas 2:32:3; a second cevian subdivides those pieces the same way.