Circle

They are equidistant from the center, They have equal lengths, They subtend equal angles at the center

Congruent chords have equal lengths by definition. From the theorem, congruent chords are equidistant from the center. Equal chords subtend equal central angles. However, congruent chords don't necessarily pass through the center (that would make them diameters) and they are not always parallel.

True

The incenter is the point where angle bisectors meet. Since angle bisectors of a triangle always meet inside the triangle (as they bisect interior angles), the incenter is always inside the triangle, regardless of whether the triangle is acute, right, or obtuse.