smc

Question

AB is a fixed diameter of a circle whose center is O. From C, any point on the circle, a chord CD is drawn perpendicular to AB. Then, as C moves over a semicircle, the bisector of angle OCD cuts the circle in a point that always:

Accepted Answer

FIGURE Draw in diameter CE. Note that, as it is inscribed in a semicircle, CDE always has a right angle at D. Since it is given that AB CD, AB DE by congruent corresponding angles. CP bisects DE, as well as the arc subtended by DE, DPE. Because AB DE, CP always bisects the arc AB.