[As always, don't forget to give proper attribution when using the following in the classroom or elsewhere as indicated in the sidebar]
The cone in the sphere problem led me to an interesting relationship in the corresponding 2-dimensional case with a surprise ending. (Only a math person would compare a math problem to a mystery novel!). The following investigation allows the student to explore a myriad of possibilities: from similar triangles to the altitude on hypotenuse theorems to Pythagorean, to chord-chord or secant-tangent power theorems, coordinate methods, draw the radius technique, etc. Sounds like this one problem might review over 50% of a geometry course? You decide for yourself! Just remember -- one person is not likely to think of every method. Open this up to student discovery and watch miracles unfold...
STUDENT ACTIVITY OR READER CHALLENGE
In the diagram above, segment AF is a diameter of the circle whose center is O, BC is a tangent segment (F is the point of tangency), BC = AF and BF = FC. Segments AB and AC intersect the circle at D and E, respectively. Lots of given there! Perhaps some unnecessary information?
(a) If AF = 40, show that DE = 32.
Notes: To encourage depth of reasoning, consider requiring teams of students to find at least two methods.
(b) Let's generalize (of course!). This time no numerical values are given. Everything else is the same. Prove, in general, that DE/BC = 4/5.
(c) So where's the 3-4-5 triangle (one similar to it, that is)? Find it and prove that it is indeed similar to a 3-4-5.