Ptolemy's Theorem The Greek astronomer Claudius Ptolemy (c. 100-170 C.E.), who invente...

Ptolemy's Theorem The Greek astronomer Claudius Ptolemy (c. 100-170 C.E.), who invented sine tables to be used for calculations in astronomy, discovered an interesting formula relating the sides and diagonals of a cyclic quadrilateral (one whose vertices lie on a circle). Use these instructions in Sketchpad to reveal this theorem.

[1] Construct ΔABC.

[2] Construct the circumcircle of ΔABC by selecting points A, B, and C and using Arc Through Three Points from the CONSTRUCT menu, then selecting points A, C, and B and repeating this command.

[3] Locate point D at random (not on the circle). Construct segments , and ,then, select each of the six segments and display their lengths on the screen. (Use Length from the MEASURE menu.)

[4] Using Calculate from the MEASURE menu, display the value

x = AB • CD + BC • AD - AC • BD (where AB = m, etc.)

Observe the value of x as you drag point D. Is this value ever negative? Would you conjecture that perhaps x ≥ 0? Is x ever zero? (Be sure to move D on arc AC) What do you think Ptolemy's Theorem might be for a cyclic quadrilateral whose sides are of length a, b, c, and d, and diagonals, m and n, as shown?

*see problem 10, section 1.4