Type three side lengths a, b, and c, or once a valid triangle exists, drag its third vertex, to explore the triangle inequality theorem: every side must be shorter than the sum of the other two. Two dashed circles show every point that is exactly distance b from one fixed end and exactly distance a from the other; where they cross is the only place the third vertex can be. If the circles do not reach each other, or one sits entirely inside the other, no triangle exists, and the gap is measured directly. An info button opens a drawer explaining the theorem.