Check the checkbox for First Chord. The illustration shows that the line segment from the center of the circle to the midpoint of the chord is part of the perpendicular bisector of the chord. The length of this line segment is the distance the chord is from the center of the circle. If you start with the line segment then you can prove that it is part of the perpendicular bisector. If you start with the perpendicular bisector, then you can prove that it contains the center of the chord. [br][br]Now ccheck Second Chord. This second chord is congruent to the first chord and it is the same distance from the center of the circle as the first chord. If you start by being given that the two chords are congruent, then you can prove that they are equidistant from the center of the circle. Conversely, if you are given that the two chords are equidistant from the center of the circle, then you can prove that the chords are congruent.[br][br]See if you can complete the details of these proofs.

Check the checkbox for Tangent Line. You will see that the tangent line is perpendicular to a radial segment going from the center of the circle to the point of tangentcy. Can you prove that if you start with a perpendicular line to the radial segment at the endpoint on the circle that it has to be a tangent line? Can you prove the converse: if you start with a tangent line, then it has to be perpendicular to this radial segment? Note that this also implies that for each point on the circle there exists exactly one tangent line to the circle through that point.