2. Euclid assumed the compass to be collapsible. That is, given two points P and

Question

2. Euclid assumed the compass to be collapsible. That is, given two points P and Q, the compass can draw a circle with center P passing through Q (Postulate III); however, the spike cannot be moved to another center 0 to draw a circle of the same radius. Once the spike is moved, the compass collapses. Being "given" a line means being given two or more points on it.) (a) Given three points P, Q, and R. Construct with a straightedge and collapsible compass a rectangle PQST with PQ as a side and such that PT is congruent to PR (b) Given a segment PQ and a ray AB. Construct the point C on AB such that PQ is congruent to AC. (Hint: Using (a), construct rectangle PAST with PT congruent to PQ, and then draw the circle centered at A and passing through S.) Exercise (b) shows that you can transfer segments with a collapsible compass and a straightedge, so you can carry out all constructions as if your compass did not collapse.

Explanation / Answer

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