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Geometry Mathley 13.1 fixed product

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June 7, 2020
fixedequal anglescirclegeometry

Problem Statement

Let ABCABC be a triangle with no right angle, EE on the line BCBC such that ∠AEB=∠BAC\angle AEB = \angle BAC and ΔA\Delta_A the perpendicular to BCBC at EE. Let the circle γ\gamma with diameter BCBC intersect BABA again at DD. For each point MM on γ\gamma (MM is distinct from BB), the line BMBM meets ΔA\Delta_A at M′M' and the line AMAM meets γ\gamma again at M′′M''. (a) Show that p(A)=AM′×DM′′p(A) = AM' \times DM'' is independent of the chosen MM. (b) Keeping B,CB,C fixed, and let AA vary. Show that p(A)d(A,ΔA)\frac{p(A)}{d(A,\Delta_A)} is independent of AA.
Michel Bataille
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