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

135. Tangents from a point; diameters of the parabola

Goal: Your student can draw the two tangents from an outside point, and prove the theorems about the chord of contact and diameters.

⏱ 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)
  • 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 V–VIII (§§ 880–892) with corollaries and the definitions of chord of contact and diameter.

The lesson, step by step

  1. Warm up

    Ask: How do you draw the tangent at a point of a parabola?

  2. The new idea

    To draw a tangent to a parabola from an external point.

    The line joining the focus to the intersection of two tangents makes equal angles with the focal radii drawn to the points of contact.

    If two tangents RP and RQ are drawn from a point R to a parabola, the line drawn through R parallel to the axis bisects the chord of contact.

    § 889: the diameters of a parabola are parallel to its axis.

  3. Draw the figure

    On your parabola, pick a point R outside. Do Prop. V: with centre R and radius RF cut the directrix at B and C; lines from B and C parallel to the axis meet the curve at the points of contact.

  4. Walk through the proof

    V: RB = RF and PB = PF, so triangles RPB and RPF are equal and RP bisects angle BPF (§ 869).

    VI: triangles RPF and RBP are equal, as are RQF and RCQ; and RB = RC, so the angles work out equal.

    VII: R is equally far from B and C, so the line through R parallel to the axis bisects BC, and so bisects PQ.

    VIII: use VII twice with the tangent at S.

  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.

  6. Wrap up

    Ask: Where do the tangents at the two ends of the latus rectum meet? (On the directrix, § 884.)

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