Wentworth's Plane and Solid Geometry · Book II, Props. IX–XI · Book 3: Book II: The Circle

25. Tangents, parallels, and the circle through three points

Goal: Your student can prove that a line perpendicular to a radius at its end is a tangent, that parallels cut off equal arcs, and that just one circle passes through three points.

⏱ About 40 minutes · 6 steps

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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 IX–XI (§§ 253–259) with corollaries.

The lesson, step by step

  1. Warm up

    Ask: Prove that a diameter perpendicular to a chord bisects it.

  2. The new idea

    A straight line perpendicular to a radius at its extremity is a tangent to the circle.

    Parallels intercept equal arcs on a circumference.

    Through three points not in a straight line one circumference, and only one, can be drawn.

  3. Draw the figure

    Draw a circle with centre O and radius OA, and the line MB through A perpendicular to OA.

    For Prop. XI: mark three points A, B, C (not in a line), draw AB and BC, and construct the perpendicular bisector of each with the compasses. Where they cross, put the compass point and draw the circle through A.

  4. Walk through the proof

    IX: any other line from O to MB is longer than the perpendicular OA (Book I, Prop. VII), so every other point of MB is outside the circle.

    X: three cases (a tangent and a secant, two secants, two tangents). In each, a diameter perpendicular to the parallels bisects the arcs, and equal pieces are added or subtracted.

    XI: the crossing point of the two perpendicular bisectors is equally far from A, B and C. It is the only such point, because two lines cross only once.

  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: Can two different circles cross in three points? (No: § 259.)

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