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There are two theorems:

  1. All three angles AC_1B, AC_2B, AC_3B are equal. In general: All angles above a chord are equal.
  2. The size of any angle above a chord AB is half the size of angle AOB where O is the center of the circle.

I am Czech. In my native language there is a name of both these two statements separately.

  1. věta o obvodovém úhlu
  2. věta o středovém úhlu

What are these two theorems called in English?

Thank you.


1 Answer 1


The inscribed angle theorem. Sometimes known as the Star Trek Lemma. A generalization of Thale's Theorem.


Wikipedia image

  • $\begingroup$ Thank you for your help. $\endgroup$
    – Jan
    Oct 4, 2021 at 11:15
  • $\begingroup$ Corollary: The unlabeled green angle (i.e. the one whose measure is π - θ) is exactly half the unlabeled center reflex angle (i.e. the one whose measure is 2π - 2θ). Therefore, you technically don't need the "above the chord" condition at all. You just need to clarify which of the two central angles you mean (i.e. when the angle is inscribed on the minor arc, you use the reflex angle, when it's inscribed on the major arc, you use the non-reflex angle). $\endgroup$
    – Kevin
    Oct 9, 2021 at 20:39

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