⁦Ceva, Menelaus, Ptolemy and the Euler line⁩

⁦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.