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| Math Central | Quandaries & Queries |
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Question from Laura, a student: I have tangents from point A and B that intersect at C. A third tangent XY lies inbetween the two lines that I have already drawn. I measured the perimeter and then I drew another line that was tangent to the circle and was inside the two lines again and measured the perimter again. The perimeters were the same but I don't know how to prove why this happened and write a theorem for it. Please help me. |
Laura,
Use the theorem that the lengths of the two tangents from a point outside a circle are equal (measured from the outside point to the point of tangency. In fact, you use the theorem three times.
Label the point T where XY is tangent the circle. In your problem you have two tangents to your circle from X: XT is a portion of XY and XA a portion of the tangent CA. Those tangents have the same length -- that is, XA = XY. Similarly, you have two tangents from Y. The length of the line XY equals the sum of those two tangents (XT + TY). Now write the perimeter, namely CX + XY + YC, using those other pieces, namely XA and YB, in place of XY.
Chris
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