14. Loci: the perpendicular bisector and the angle bisector

The lesson, step by step

1. Warm up

   Ask: State the hypotenuse-leg theorem (Prop. XXV).

2. The new idea

   Read §§ 156–158. A locus is the line (or lines) containing all the points that satisfy a condition, and no others.

   The perpendicular bisector of a given line is the locus of points equidistant from the extremities of the line.

   The bisector of a given angle is the locus of points equidistant from the sides of the angle.

   To prove a locus you show two things: every point on the line works, and no point off it works.

3. Draw the figure

   Draw a segment AB. With the compasses, find several points each the same distance from A and B (equal arcs from A and from B). Your student will see they line up on the perpendicular bisector.

   Then draw an angle and a point O inside it with perpendiculars OF and OG to the two sides.

4. Walk through the proof

   XXX, part 1: for O on the bisector, the right triangles OPA and OPB have equal legs, so OA = OB.

   Part 2: for C off the line, CA crosses the bisector at D; DA = DB, and CB < CD + DB = CA.

   XXXI: if O is equally far from the sides, the right triangles AOF and AOG are equal (hypotenuse and leg), so AO bisects the angle. Conversely, a point D on the bisector gives two right triangles equal by § 141.

5. Your student proves it back

   Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.

   If a step is missing, ask "How do we know that?" rather than supplying it.

6. Wrap up

   Ask: Where are all the points in a field that are the same distance from two trees? (On the perpendicular bisector of the line joining them.)