The theorem of Apollonius by dissection

Abstract

The figure shews the dissection in the formLet m, x denote AD, DE, where E is the projection of A on BC.Quadrilateral 1 is constructed with sides ½ (c + x), ½ (c – x), ½ (m + ½a + x), ½ (m – ½a – x); quadrilateral 6 with sides ½ (b – x), ½ (b + x), ½ (m – ½a – x), ½ (m + ½a – x). The square on AB is then dissected into four quadrilaterals equal to 1, plus the shaded square (AD2); similarly for the square on AC. These quadrilaterals are reassembled in the squares on BE, EC respectively with the right-angled corners in the reversed positions, so as to enclose squares each equal to DE2. Finally two quadrilaterals are subdivided (4, 11) and (3, 5, 10, 12) to fit in as shewn in the figure.