Question

which of the following is true about the incenter of a triangle?

it is the point that is equidistant from each of the altitudes.
it is the point that is equidistant from each vertex.
it is the point that is equidistant from each of midpoints.
it is the point that is equidistant from each side of the triangle.

Explanation:

Define the incenter of a triangle

The incenter is the intersection point of the three angle bisectors of a triangle.

Apply the Incenter Theorem

By the Incenter Theorem, the incenter is equidistant from all three sides of the triangle, serving as the center of the inscribed circle (incircle).

Evaluate the given options

• Option 1: Incorrect. The orthocenter is related to altitudes, but not equidistant from them.
• Option 2: Incorrect. The circumcenter is equidistant from each vertex.
• Option 3: Incorrect. No standard triangle center is defined as equidistant from all midpoints.
• Option 4: Correct. The incenter is equidistant from each side of the triangle.

Answer:

• It is the point that is equidistant from each of the altitudes.
• It is the point that is equidistant from each vertex.
• It is the point that is equidistant from each of midpoints.
• It is the point that is equidistant from each side of the triangle. (Correct answer)