Теория:

Theorem \(1\). Each point of the bisector of an undeveloped angle is equidistant from its sides.
 
Theorem \(2\). (Converse) A point lying inside an undeveloped angle and equidistant from its sides lies on the bisector of this angle.
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Theorem \(3\). Each point of the middle perpendicular to the segment is equidistant from its ends.
 
Theorem \(4\). (Inverse) A point equidistant from the ends of the segment lies on the middle perpendicular to it.
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Important!
The first remarkable point of the triangle is the intersection point of the bisectors.
Theorem \(5\). The bisectors of a triangle intersect at one point.
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In the above picture \(AN\), \(BM\) is the bisectors, \(O\) is their intersection point.
Is the bisector \(CK\)? If the point \(O\) is equidistant from the sides \(AB\) and \(AC\)  and the sides \(BA\) and \(BC\), then it lies on the bisector of the angle since it is equidistant from the sides of the corner.
This point is the center of the circle inscribed in the triangle, always in the triangle.
 
Important!
The second remarkable point of the triangle is the intersection point of the middle perpendiculars of the sides of the triangle.
Theorem \(6\). The middle perpendiculars to the sides of the triangle intersect at one point.
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  • Suppose in the above figure; that the point \(O\) is the intersection point of the two middle perpendiculars of the sides \(AB\)  and \(BC\). It is equidistant from the points \(A\) and \(B\), and from the points \(B\) and \(C\). Therefore, it lies on the middle perpendicular of the side \(AC\), since it is equidistant from its endpoints. \(O\) is the intersection point of the two middle perpendiculars of sides \(AB\) and \(BC\). And it is equidistant from the points \(A\) and \(B\), and from the points \(B\) and \(C\). Therefore, it lies on the mid-perpendicular of the side \(AC\), as it is equidistant from its endpoints \(A\), \(B\), \(C\).
  • This point is the center of the circle circumscribed around the triangle, located in triangles with sharp corners, outside a triangle with an obtuse angle and on the hypotenuse of a right triangle.
Circle circumscribed around a triangle
 A circle is called circumscribed around a triangle if all the vertices of the triangle are located on a circle.
Its center is equidistant from all vertices, that is, it should be at the intersection of the middle perpendiculars to the sides of the triangle.
Consequently, a circle can be described around any triangle, since the middle perpendiculars to the sides intersect at one point.
 
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