Angle and Circle: Some Background Tools for Alice's Adventures

As you read Journey ... you probably won't be able to verify all the statements made nor solve all the problems, but there are a few basic ideas that will help you with some of the reasoning involved. In this lab you'll make some constructions and draw (metaphorically?) some conclusions as background for the reading in Journey .... I hope they'll be interesting in themselves, as well!

Many of the constructions you make will be dynamic versions of diagrams in Journey ...; each should indicate in a dynamic way the infinitely many ways that a diagram in the text might have been drawn to illustrate the same principles.

As a final note before starting, the conclusions will involve relying on the 5th Postulate, so they are Euclidean results, not results of absolute geometry.

Constructions for the Diagrams on Page 13:

1. Construction 1: Construct a circle with O as center, a diameter AB, and any other point C on the circle. The circle and points should be constructed so that neither A nor B is the "size control point" of the circle, A and C can be dragged anywhere on the circle (and always remain on the circle) without the size of the circle changing, and AB should always remain a diameter. Be sure to test your construction for "unmess-upability."

2. Investigation 1: By using Proposition 5 (Equal sides implies equal base angles) and Proposition 32 (Interior angle sum is two right angles) of Euclid, determine a relationship between angle BAC and angle BOC in Construction 1.

State the relationship:

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3. Investigation 2: By using Proposition 5 and Proposition 32 again, determine a relationship between angle BAC and angle BCO in Construction 1.

State the relationship:

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4. Based on the relationship in number 3 above and Proposition 32, what can be said about an angle inscribed in a circle if the angle subtends a diameter? (Meanings of the two italicized terms are illustrated in Diagram 2 below, where the acute angle opposite the diameter is inscribed in the circle and subtends that diameter.)

Diagram 2

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5. Construction 2: Construct a circle with O as center, a chord AE, any other point D on the circle, and the tangent to the circle at point A. As in #1, the circle and points should be constructed so that the "size control point" of the circle is hidden, and A, D, and E can be dragged anywhere on the circle (and always remain on the circle).

6. Investigation: Observe what happens to angle ADE as point D is dragged around the circle. How does the angle change?

Optional: Make additions to your diagram so that it resembles the diagram below, and compare angles ADE, AGE, and AHE. Then prove that no matter where D is on the larger arc subtended by chord EA on the circle, the angle inscribed at D and subtended by AE is the same. (Thanks to Laura Barringer for clarifying the assumption concerning the possible locations for point D.) What happens if D is on the smaller arc subtended by chord EA?

Some elements of one method of proof:

Optional: Show that the angle FAE, formed by the tangent AF to the circle at point A, is also equal to angle EDA.