Construct the locus of points P such that angle APB = angle ACB, given triangle ABC.

• Place the compass needle at point B and open it to a radius greater than half the length of BC. Draw an arc above and below BC.
• Without changing the compass radius, place the needle at point C and draw arcs above and below BC, intersecting the previously drawn arcs.
• Draw a straight line connecting the two points where the arcs intersect. This line is the perpendicular bisector of BC.

The condition $\angle APB = \angle ACB$ means that P lies on a circle that passes through points A, B, and C. This is the circumcircle of $\triangle ABC$.

• To find the center of this circle (the circumcenter), construct the perpendicular bisector of another side, for example, AB.
  • Place the compass needle at point A and open it to a radius greater than half the length of AB. Draw an arc above and below AB.
  • Without changing the compass radius, place the needle at point B and draw arcs above and below AB, intersecting the previously drawn arcs.
  • Draw a straight line connecting the two points where these new arcs intersect. This line is the perpendicular bisector of AB.
• The point where the perpendicular bisector of BC (from Step 1) and the perpendicular bisector of AB intersect is the circumcenter of $\triangle ABC$. Let's call this point O.
• Place the compass needle at O and open it to the radius OA (or OB or OC, as they are all equal).
• Draw the circle. This is the circumcircle of $\triangle ABC$.
• The locus of point P is the arc of this circumcircle that lies on the same side of AB as point C.