Describe the locus of all points in a plane that are equidistant from the sides of a triangle.

Question

1. A single point that is the center of the inscribed circle of the triangle 

2. A triangle similar to but smaller than the original triangle 

3. A triangle similar to but larger than the original triangle

4. A single point that is the center of the circumscribed circle of the triangle.

Answer 1

The locus is a single point called the in center.   The in center is the point where all three angle bisectors intersect. 

The in center is the center of the inscribed circle.  The inscribed circle is tangent to all three sides.