The inscribed angle theorem states that ∠ O = 2 × ∠ B. The theorem is true for when point B is located between points A and C relative to the perimeter. But what would happen if B was located exactly at A or C? Angle B would obviously equal 0 and so would the length of one of it's legs, but my question is, is this transition discrete or continuous? In other words, as the length of either line BA or BC approaches 0, does angle B also approach 0? Or is it unaffected?

Question

The inscribed angle theorem states that   ∠  O  =  2  ×  ∠  B. The theorem is true for when point B is located between points A and C relative to the perimeter. But what would happen if B was located exactly at A or C? Angle B would obviously equal 0 and so would the length of one of it&#039;s legs, but my question is, is this transition discrete or continuous? In other words, as the length of either line BA or BC approaches 0, does angle B also approach 0? Or is it unaffected?

hottchevymanzm · Accepted Answer

Step 1First, I don&#039;t think you are correct when saying &quot;      ∠    B would obviously equal 0&quot; - this is like saying that an angle of a triangle that is incident with a side of length 0 is 0 while it is, by law of cosines, more accurately described as   a  r  c  c  o  s  (      1    0    ) which is clearly undefined.Step 2Second, a pre-condition of the theorem is that AB and BC are chords of the circle, which is not true if   B  =  A (or   B  =  C). In all other cases, the theorem will hold. In particular, when BA or BC approaches 0, the angle at B will not approach 0 but stay equal to                     ∠            O        2  , then &quot;jump&quot; to being undefined when AB (or BC) reaches 0.

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