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

42. A line parallel to a side of a triangle

Goal: Your student can prove that a parallel to one side divides the other two proportionally (both cases), and the converse.

⏱ About 40 minutes · 7 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 XIII and XIV (§§ 342–347) with corollaries, internal and external division, and Exercises 246–248.

The lesson, step by step

  1. Warm up

    Ask: State "composition" and "the series of equal ratios" theorems.

  2. The new idea

    If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally.

    If a straight line divides two sides of a triangle proportionally, it is parallel to the third side.

  3. Draw the figure

    Draw a large triangle ABC. Draw EF parallel to BC, cutting AB at E and AC at F. Measure AE, EB, AF and FC and compute EB : AE and FC : AF. They should nearly agree.

  4. Walk through the proof

    XIII, Case 1 (commensurable): a common measure fits m times in EB and n times in AE. Parallels through the division points cut AC into equal parts too (§ 187), m in FC and n in AF.

    Case 2 (incommensurable): as in Book II, Prop. XVI, cut off a commensurable part and let the leftover shrink to zero; equal ratios have equal limits.

    XIV: draw EH parallel to BC. Then AC : AF = AC : AH, so AF = AH, and EF coincides with EH.

  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

    Ex. 246. Find the fourth proportional to 91, 65, and 133.

    Ex. 248. Find the third proportional to 54 and 3.

    Then Ex. 247 (the mean proportional between 39 and 361).

  7. Wrap up

    Read §§ 346–347: dividing a line internally and externally in the same ratio (harmonic division).

Answers to the book's problems

Ex. 246: 95. Ex. 247: √(39 × 361) = 19√39, about 118.65. Ex. 248: 1/6.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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