Hint for Exercise 2.2. The Compass Equivalence Theorem
It has been said that figures rule the world. Maybe. But I am sure that figures show us whether it is being ruled well or badly.
—Johann Wolfgang von Goethe (1749–1832)

Drag the red points to explore the construction.
To reset to the original settings type the letter "R" on the keyboard.

Hint 1. Drag A close to B.
Hint 2. Drag r to make circles centered at A with radius AP and centered at B with radius r different sizes.
Hint 3. Drag B or r so that they are on opposite sides of A.
You should find that the proof is not valid in at least two ways.

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The Geometer's Sketchpad.