MathDB
Locus of Angle Bisector on a Semicircle

Source:

February 1, 2009
geometryperpendicular bisectorangle bisector

Problem Statement

AB AB is a fixed diameter of a circle whose center is O O. From C C, any point on the circle, a chord CD CD is drawn perpendicular to AB AB. Then, as C C moves over a semicircle, the bisector of angle OCD OCD cuts the circle in a point that always: <spanclass=latexbold>(A)</span> bisects the arc AB<spanclass=latexbold>(B)</span> trisects the arc AB<spanclass=latexbold>(C)</span> varies <span class='latex-bold'>(A)</span>\ \text{bisects the arc } AB \qquad<span class='latex-bold'>(B)</span>\ \text{trisects the arc } AB \qquad<span class='latex-bold'>(C)</span>\ \text{varies} <spanclass=latexbold>(D)</span> is as far from AB as from D<spanclass=latexbold>(E)</span> is equidistant from B and C <span class='latex-bold'>(D)</span>\ \text{is as far from }AB \text{ as from } D \qquad<span class='latex-bold'>(E)</span>\ \text{is equidistant from }B \text{ and } C