Converse Perpendicular Chord Bisector Theorem

It is given that QS is a perpendicular bisector of TR. Let L be the center of the circle. Consider △ LPT and △ LPR.

Segments LT and LR are radii of ⊙ L. Since all radii of a circle are congruent, LT and LR are congruent. Additionally, by the definition of a perpendicular bisector, it can be stated TP and RP are congruent segments. Furthermore, by the Reflexive Property of Congruence, it can be said that LP is congruent to itself. LT≅ LR TP≅ RP LP≅ LP This information can be seen in the diagram.

Here, three sides of △ LPT are congruent to three sides of △ LPR. Therefore, by the Side-Side-Side Congruence Theorem, △ LPT and △ LPR are congruent triangles. Since corresponding parts of congruent triangles are congruent, ∠ LPT and ∠ LPR are congruent angles. △ LPT ≅ △ LPR ⇓ ∠ LPT ≅ ∠ LPR By the definition of a chord, ∠ TPR is a straight angle. By dividing 180^(∘) by 2, it can be calculated that ∠ LPT and ∠ LPR measure 90^(∘) each. m∠ LPT=90^(∘) and m∠ LPR=90^(∘) This means that they are right angles and that LP is perpendicular to TR. Therefore, LP is the perpendicular bisector of RT. Since there is only one perpendicular bisector that can be drawn to a segment, L must lie on QS.

Finally, the fact that QS contains the center of the circle leads to the conclusion that QS is a diameter of ⊙ L.