Wentworth's Plane and Solid Geometry · Book III, Exs. 267–285 · Book 4: Book III: Proportion. Similar Polygons
51. Original theorems on lines in circles and triangles
Goal: Your student proves original theorems using the numerical properties of figures.
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
Pages 170–171: theorem exercises 267–285, with many hints that cite §§ 371, 377, 378 and 381.
The lesson, step by step
Warm up
Ask: State the theorems on two chords, and on a tangent and a secant.
Together
Ex. 268. The tangents to two intersecting circles drawn from any point in their common chord produced, are equal. (§ 381.)
Each tangent squared equals the same product of the secant along the common chord.
On their own
Your student picks two or three: Ex. 267, 269, 270, 273 or 277.
Ex. 270. If three circles intersect one another, the common chords all pass through the same point.
A famous result
Read Ex. 280 and its hint. Draw a quadrilateral and check it with a ruler before proving it.
Wrap up
Your student explains which section each proof depended on.
Answers to the book's problems
Ex. 268: each tangent squared equals the product of the whole secant through the common chord and its external part (§ 382), the same for both circles. Ex. 269: by Ex. 268 the two tangent pieces from the meeting point are equal. Ex. 277: the radius to the point of contact is the altitude of a right triangle (the angle at the centre is right), so it is the mean proportional between the segments (§ 367).
Worked out for this site from the scan. If a number in your copy differs, trust the book.