For each property, pick the point of concurrency that it applies to

the points are Circumcenter Incenter
questions
Contained in the perpendicular bisectors of a triangle

Equidistant from the vertices of a triangle

Contained in the angle bisectors of a triangle

Equidistant from the sides of a triangle

Contained in the perpendicular bisectors of a triangle: Circumcenter

Equidistant from the vertices of a triangle: Circumcenter

Contained in the angle bisectors of a triangle: Incenter

Equidistant from the sides of a triangle: Incenter