Posts tagged 'median'

F.35 Obtuse and Acute Triangles

By Euclid’s Elements II.12 (excess) and II.13 (deficit), the square on side a opposite an obtuse or acute angle satisfies a^2 = b^2 + c^2 +/- 2 c x, where x = |a^2 - (b^2 + c^2)| / (2 c) is the projection of side b onto line c and h = sqrt(b^2 - x^2) is the altitude.

F.46 Stewart’s Theorem

If a cevian of length f from C divides base c = AB into segments k’ c and k c (with k + k’ = 1), Stewart’s Theorem states f^2 = k a^2 + k’ b^2 - k k’ c^2; at the midpoint k = k’ = 1/2, this yields Apollonius’s median formula f^2 = (1/2)(a^2 + b^2 - c^2/2).