Bisector (apex)
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Every point on the bisector of an angle is equidistant from the sides of that angle. It is understood that the distance of a point from a line is the length of the perpendicular dropped from the point to the line.
Not always but yes if the shape is a square
Angle bisectors are.
(The set of all points equidistant from the two sides of the angle. :)
Incentre.