Circles, Tangent Lines, & Triangles Hint #2

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

The diagram below contains two final additions, which will help you to complete the logical chain of facts. Study the diagram and try to visualize how these additions can help you.

The radius OC has been drawn in, to remind you that the tangent line is perpendicular to the radius of the circle at the tangent point, C. Also drawn in is the semicircle with diameter BH and center at point T. How do you know that the center is point T? By reversing the previous argument about the tangent line and the radius. OC is perpendicular to TC. Since angle BCH is a right angle, a semicircle can be constructed that passes through point C and has diameter BH. Therefore, OC is tangent to this semicircle at point C, and TC must be a radius of the semicircle (since it is perpendicular to the tangent line at point C). Therefore, TC = TB = TH, since all are radii of the semicircle.

If you've done your background research, you should be able to complete the proof with the information you now have available.

Return to Hint #1.

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