Ceva, Menelaus, Ptolemy and the Euler line
Optional stories from Grade 8 geometry: when three cevians meet at a point, when three points are collinear, the product of the diagonals of a cyclic quadrilateral, and the Euler line.
- Ukraine: grade 8
- Russia: grade 9
Guide
These theorems are in the Grade 8 geometry textbook as optional reading. The segments AA₁, BB₁, CC₁ joining the vertices of triangle ABC to points A₁, B₁, C₁ on the opposite sides are called cevians.
Ceva's theorem. The cevians AA₁, BB₁, CC₁ meet at one point if and only if (AC₁ : C₁B) · (BA₁ : A₁C) · (CB₁ : B₁A) = 1. Menelaus' theorem: points A₁, B₁, C₁, two on the sides and one on the extension of a side, are collinear if and only if the same equality holds.
Ptolemy's theorem. For a quadrilateral inscribed in a circle, the product of the diagonals equals the sum of the products of opposite sides: AB · DC + BC · AD = BD · AC.
The Euler line. In every triangle the circumcentre O, the centroid M and the orthocentre H lie on one line, and HM = 2MO.