Wentworth's Plane and Solid Geometry · Book IX, Props. XX–XXII · Book 10: Book IX: Conic Sections

141. Tangents at corresponding points; conjugate diameters

Goal: Your student can prove that tangents at corresponding points meet the axis at the same point, and the theorems on conjugate diameters.

⏱ About 40 minutes · 6 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)
  • Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)

What's in the book

Propositions XX–XXII (§§ 931–941) with corollaries and definitions of normal, subtangent, subnormal and conjugate diameters.

The lesson, step by step

  1. Warm up

    Ask: What is the auxiliary circle? What are corresponding points?

  2. The new idea

    The tangents drawn at two corresponding points of an ellipse and its auxiliary circle cut the major axis produced at the same point.

    The tangents drawn at the ends of any diameter are parallel to each other.

    If one diameter is conjugate to a second, the second is conjugate to the first.

  3. Draw the figure

    Draw an ellipse with its auxiliary circle. At a point Q of the circle draw the tangent to T on the axis; join T to the corresponding point P of the ellipse.

  4. Walk through the proof

    XX: through any other point S of the ellipse draw a perpendicular to the axis, meeting TP at R, the circle at K and the circle's tangent at L. Then RB : LB = SB : KB = b : a, and LB > KB, so RB > SB: R is outside the ellipse.

    § 932: OT × OM = a².

    XXI: the ends of a diameter have equal abscissas of opposite sign, so their tangents meet the axis at equal distances on either side; the triangles are equal.

    XXII: the radii OQ and OS of the auxiliary circle at the corresponding points are at right angles; use similar triangles.

  5. Your student proves it back

    Close the book. Your student draws the curve and 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. Point to each line as it is named.

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

    These proofs are long; it is enough that your student can explain the main step of each.

  6. Wrap up

    Ask: Name a pair of conjugate diameters you can see at once. (The two axes.)

@teddyrookbook GitHub