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I know this is probably simple but I just can't see it.

How can I determine the scale factor of a pantograph from the ratio of the arms?

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The scale factor is the ratio $$\frac{\text{distance from the fixed point to the drawing point}}{\text{distance from the fixed point to the tracing point}}.$$ The mechanics of the device serve to keep that ratio constant as the mechanism is rotated, expanded, and/or contracted.

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This page breaks down the triangles pretty well.

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