ABCD is a cyclic quadrilateral.[br]a) A and A' are the inversion points of each other. Prove that [math]\Delta DT_AA\sim\Delta DA'T_A[/math] and [math]DA\cdot DA'=r^2[/math][br]b) Applying result of a) on B and C (their corresponding inversion points are B' and C' with the same inversion circle), prove that [math]\Delta DC'B'\sim\Delta DBC[/math] and express B'C' in terms of r and sides of [math]\Delta DBC[/math].[br]c) Applying the above results to another two triangles, prove the Ptolemy Theorem.