The following sequence of problems provides a proof of the validity of the Simson Line for...

The following sequence of problems provides a proof of the validity of the Simson Line for a triangle. Each problem independently contains a useful, signifi-t cant geometric result.

Ptolemy's Theorem If the sides of a cyclic quadrilateral are of length a, b, c, and d and the diagonals have lengths m and n, as shown in the figure that follows, then

Ac+bd =mn

This is proven using similar triangles, as follows (you are to fill in the details).

(1) Construct line so that ∠ABE≌∠CBD. (See figure below.)

(2) m∠EAB = m∠CDB.

(3) ΔABE ~ ΔCBD → a/m - x/c, or ac = mx.

(4) ΔABD ~ ΔEBC→ m/b = d/y, or bd = my.

(5) ∴ ac + bd = mn.