Question

choose all of the true statements that apply to the incenter of a triangle.

it can be located outside of the triangle.
it is equidistant from each vertex.
it is the intersection of the angle bisectors.
it is equidistant from each of the sides of the triangle.
it is the intersection of the three altitudes of the triangle.

Explanation:

Define the incenter of a triangle

Using the Angle Bisector Theorem knowledge point
The incenter is defined as the point where the three internal angle bisectors of a triangle intersect.

Analyze distance properties of the incenter

Using the Incenter Theorem knowledge point
The incenter is the center of the inscribed circle (incircle) of the triangle, which makes it equidistant from all three sides of the triangle.

Evaluate location and vertex distance properties

The incenter always lies strictly inside the triangle because the angle bisectors are interior rays. The circumcenter, not the incenter, is equidistant from each vertex.

Evaluate altitude intersection

The intersection of the three altitudes of a triangle is the orthocenter, not the incenter.

Answer:

• It can be located outside of the triangle.
• It is equidistant from each vertex.
• It is the intersection of the angle bisectors. (Correct answer)
• It is equidistant from each of the sides of the triangle. (Correct answer)
• It is the intersection of the three altitudes of the triangle.