Wentworth's Plane and Solid Geometry · Book III, Exs. 256–265 · Book 4: Book III: Proportion. Similar Polygons
47. Original theorems on similar figures
Goal: Your student proves original theorems using similar triangles.
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)
- A protractor
What's in the book
Page 159: theorem exercises 256–265 on similar triangles and proportional lines, some with hints.
The lesson, step by step
Warm up
Ask: List the three tests for similar triangles.
The usual plan
To prove a product of lines equal to another product, find two triangles whose sides include those lines, prove them similar, write the proportion, then multiply extremes and means.
Together
Ex. 265. The bisector of the vertical angle A of the triangle ABC intersects the base at D and the circumference of the circumscribed circle at E. Show that AB × AC = AD × AE.
Hint: triangles ABD and AEC are similar (angles B and E stand on the same arc).
On their own
Your student picks two: Ex. 257, 258, 261 or 263 are good. Ex. 263 is a famous result:
Ex. 263. In an inscribed quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides.
Wrap up
Your student shows one finished proof and names the pair of similar triangles used.
Answers to the book's problems
Ex. 258: if AD and BE are the altitudes on BC and AC, the right triangles ADC and BEC share the angle C, so they are similar (§ 356) and AD : BE = AC : BC. Ex. 264: base angles are equal, being halves of equal supplements; § 355. Ex. 265: triangles ABD and AEC are similar (§ 355), so AB : AE = AD : AC.
Worked out for this site from the scan. If a number in your copy differs, trust the book.