Circling Around

Problem Statement

    A is a fixed point on a given circle, and B is a variable point on the circle; it travels around the circle. M is the midpoint of AB. What is the locus of M?

Questions that can help you find the locus

AB is a chord of the circle with centre O, M is the mid-point of the chord. What is angle OMB? Will the value of angle OMB change as B moves around the circle? Experiment with GeoGebra and make a conjecture about angle OMB. Justify your conjecture.

(iii)                 Can you now predict the locus of M?
          Move the point B around the circle, then redefine your conjecture if you want. Click on the check box to see the locus. Is this the locus you predicted?

Justify your stance. Hint: You will need to know a theorem about the angle in a semi-circle.