Which circle has the three sides of a triangle as tangents?

• A. Excircle
• B. Circumcircle
• C. Centroid
• D. Incircle

Answer: (D)

The circle that touches every side of a triangle is the incircle. It sits inside the triangle and is tangent to each of the three sides at exactly one point. The center of this circle is the incenter, where the angle bisectors meet, and the radius (the inradius) is the common distance from the incenter to each side.

Why this one fits: being tangent to all three sides means the circle must lie inside the triangle and touch each side exactly once, which is exactly what the incircle does.

The other options don’t fit that description. An excircle is outside and touches one side plus the extensions of the other two, not all three sides. A circumcircle goes through the triangle’s vertices, not tangent to its sides. The centroid is a point related to balance, not a circle.