Final answer:
The center of the inscribed circle in a triangle, known as the incenter, lies equidistant from each side of the triangle. It is the point where the angle bisectors of all three angles intersect and is the center of the circle inscribed within the triangle.
Step-by-step explanation:
The correct answer to the question of which is always true about the center of the inscribed circle of a triangle is: d) It lies equidistant from each side of the triangle.
The center of the inscribed circle, also known as the incenter, is the point where the angle bisectors of the triangle intersect. The incenter is equally distant from all sides of the triangle because it is the center of the circle that is inscribed within the triangle, touching each side at exactly one point. This circle is called the incircle. The distance from the incenter to any point of the incircle is called the inradius. The concept of distance from a point to a line refers to the shortest distance, which is always the perpendicular from the point to the line (side of the triangle in this case).
Option a) is incorrect because the incenter is equidistant from the sides of the triangle, not the vertices. Option b) is incorrect because the incenter is not necessarily on the perpendicular bisector of one of the sides but lies at the intersection of all the angle bisectors. Option c) is incorrect because the incenter doesn't necessarily lie at the midpoint of the longest side, as the location of the incenter is based on the angle bisectors, not the lengths of the sides.