32. [ 18 ] Circle Ω has radius 13. Circle ω has radius 14 and its center P lies on the boundary of circle Ω. Points A and B lie on Ω such that chord AB has length 24 and is tangent to ω at point T. Find AT · BT. Answer: 56 Solution: Let M be the midpoint of chord AB ; then AM = BM = 12 and Pythagoras on triangle AM O gives M O = 5 . Note that ∠ AOM = ∠ AOB/ 2 = ∠ AP B = ∠ AP T + ∠ T P B or tan ( ∠ AOM ) = tan ( ∠ AP T + ∠ T P B ). Applying the tangent addition formula, AT BT

AM T P T P

AT BT M O 1 − · T P T P AB · T P = , 2 T P − AT · BT 2 2 from which AT · BT = T P − AB · T P · M O/AM = 14 − 24 · 14 · 5 / 12 = 56 .