The diagram below shows a triangle ABC.

1. Construct equal arcs from $A$ and $C$ and draw the bisector line.

Set the compass to a radius greater than half of $AC$. With centre $A$, draw arcs above and below $AC$. Using the same radius and centre $C$, draw arcs to intersect the first pair of arcs. Join the two intersection points with a straight line. This line is the perpendicular bisector of $AC$.

1. Construct the angle bisector of $\angle BCA$ using arcs.

With centre $C$, draw an arc cutting $CB$ and $CA$ at two points. Using these two points as centres and the same radius, draw two arcs that intersect inside $\angle BCA$. Draw a straight line from $C$ to this intersection point. This line is the bisector of $\angle BCA$.