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Circles, Tangent Lines, & Triangles Hint #1

If you're having a hard time coming up with a solution, keep in mind the strategy for completing a proof.

  1. Do your background research. The Terms, Concepts and Questions section is a good place to start!
  2. Organize your known facts (make a list!) Your list should include:
    1. the information given in the statement of the problem,
    2. relevant information from your background research, and
    3. relevant information from your knowledge of geometry.
  3. Make sure you also write down a statement of the desired solution.
  4. Try to build a list of the statements you need to prove in order to solve the problem. Remember that the goal of a proof is to construct a logical chain of steps leading from the given facts to the desired solution. Each step must be justified.
  5. Constructing the proof does not have to be a one-way process, from beginning to end. You can also build backwards from the desired solution, and have your steps meet in the middle.
  6. In addition to thinking logically, think visually!
    1. Remember that some of the facts you know about the problem will not be included in the original diagram which poses the problem. Finding ways to incorporate your known facts into the diagram may help you solve the problem.
    2. With many proofs, you also need to use your knowledge of geometry to build additional information into the diagram to solve the problem.
    3. Get yourself a few sheets of blank paper and try out your ideas as sketches.
  7. Spend some time thinking about the problem and you should be able to come up with the proof.

The diagram below contains one additional point and two line segments, which will help to build the logical chain of facts. A line has been extended from A, through C and on to the new point, H, on the tangent line through B. Additionally, a line connects points B and C. Study the diagram and try to visualize how these additions can help you.

Notice that the added line segments meet in a right angle. They also define three right triangles: ADC, BCH and ABH. If you've done your background research, this should take you most of the way through the proof.

Still stuck? Try on your own first, but if you find that you need a hint, click Hint #2.

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