Wentworth's Plane and Solid Geometry · Book III, Props. XXXII–XXXV · Book 4: Book III: Proportion. Similar Polygons

50. Chords, secants, tangents and bisectors

Goal: Your student can prove the theorems on intersecting chords, on a secant and tangent from a point, on the bisector of an angle, and on the product of two sides.

⏱ About 40 minutes · 7 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to these pages
  • A ruler (straightedge) and a pair of compasses
  • Sharp pencil, eraser and plain paper (or a geometry notebook)

What's in the book

Propositions XXXII–XXXV (§§ 378–384), and Exercise 266.

The lesson, step by step

  1. Warm up

    Ask: State Prop. XXX.

  2. The new idea

    If two chords intersect in a circle, the product of the segments of one is equal to the product of the segments of the other.

    If from a point without a circle a secant and a tangent are drawn, the tangent is the mean proportional between the whole secant and its external segment.

    In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the altitude upon the third side.

    Prop. XXXIV (§ 383) gives the square of the bisector of an angle of a triangle.

  3. Draw the figure

    Draw a circle with two chords MN and PQ crossing at O. Join MP and NQ. In a second circle, draw from an outside point A a tangent AD and a secant ABC, and join DB and DC.

  4. Walk through the proof

    XXXII: triangles NOQ and POM have equal angles (inscribed angles on the same arcs), so they are similar; write the proportion and multiply.

    XXXIII: triangles ADC and ABD share angle A, and angle ACD equals angle ADB (each measured by half arc BD). So AC : AD = AD : AB.

    XXXIV: circumscribe a circle, extend the bisector to the circle, and use similar triangles with XXXII.

    XXXV: draw the diameter from N; the right triangles NOM and NQP are similar.

  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. Exercises

    Check XXXII with numbers by measuring OM, ON, OQ and OP in your drawing and multiplying.

  7. Wrap up

    Read the notes after Props. XXXIV and XXXV: these theorems let us compute bisectors and the circumscribed radius from the three sides.

@teddyrookbook GitHub