Wentworth's Plane and Solid Geometry · Book II, Exs. 118–133 · Book 3: Book II: The Circle
31. More theorems on circles
Goal: Your student proves original theorems about tangent and intersecting circles.
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
Pages 110–111: theorem exercises 118–133, many with hints, on tangent circles, common tangents and inscribed figures.
The lesson, step by step
Warm up
Ask: What is true of the line of centres of two tangent circles?
Together
Ex. 122. Through one of the points of intersection of two circles a diameter of each circle is drawn. Prove that the line joining the ends of the diameters passes through the other point of intersection.
The book's hint is that each angle at B is 90°, an angle in a semicircle.
On their own
Your student picks two or three: Ex. 118, 119, 123, 124 or 131 are good choices.
Ex. 131. If ABC is an inscribed equilateral triangle and P is any point in the arc BC, then PA = PB + PC.
Draw to believe
Before proving Ex. 131, draw it carefully and measure PA, PB and PC with a ruler. Then follow the hint: take PM = PB on PA.
Wrap up
Your student says which exercise was hardest and what idea unlocked it.
Answers to the book's problems
Ex. 122: the angles at the second point of intersection, standing on the two diameters, are each right angles (§ 290), so together they make a straight line. Ex. 124: let the tangent lengths from the two acute vertices be x and y; then the legs are x + r and y + r and the hypotenuse is x + y, so the legs together exceed the hypotenuse by 2r. Ex. 131: triangle PBM is equilateral, and triangles ABM and CBP are equal (§ 143), so AM = PC.
Worked out for this site from the scan. If a number in your copy differs, trust the book.