ABC is acute-angled. The circle diameter AB meets the altitude from C at P and Q. The circle diameter AC meets the altitude from B at R and S. Show that P, Q, R and S lie on a circle.
Solution
Use vectors. Take A as the origin. Let AB = b, AC = c, AR = r. AR is perpendicular to RC, so r.(c - r) = 0. BR is perpendicular to AC, so (b - r).c = 0. Hence r.r = r.c = b.c. Thus |AR| = √(|AB|·|AC| cos A). But the identical argument gives the same value for |AS|. The situtation is symmetrical between B and C, so we get the same result for |AP| and |AQ|. Hence all four points lie on a circle center A.
© John Scholes
jscholes@kalva.demon.co.uk
11 May 2002