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I looking for a name of this theorem or lemma,

"In given triangle ABC, the angle bisector of angle BAC meets with a circumcenter at middle point of arc BC".

I can't find this theorem or lemma's name in English.

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    $\begingroup$ You presumably meant "circumcircle, not "circumcenter". I doubt that this result has a standard name because it's an immediate corollary of the fact that the measure of an angle inscribed in a circle is half the measure of the arc it subtends. $\endgroup$ – Andreas Blass Mar 22 '17 at 16:04
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The theorem that equal angles at the circumference of a circle stand on equal arcs is Euclid III,26. Hence the bisector of $\angle BAC$ bisects arc $BC$. It will pass through the circumcenter, however, only if $AB=AC$, since then it will be the perpendicular bisector of $BC$ (see Euclid III,1 Porism; VI,3).

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