Enter the m2p object snap. This will find the midpoint between any two picked points without the need for a construction line.
How it works: Say you want to draw a circle midway between two squares, start the circle command and then enter "m2p" when prompted to specify the center point. Now just follow the prompts.
Command: circle
Specify center point for circle or [3P/2P/Ttr (tan tan radius)]: m2p
First point of mid: (pick first point)
Second point of mid: (pick second point)
Specify radius of circle or [Diameter] <20.000>: (pick to complete circle)
Specify center point for circle or [3P/2P/Ttr (tan tan radius)]: m2p
First point of mid: (pick first point)
Second point of mid: (pick second point)
Specify radius of circle or [Diameter] <20.000>: (pick to complete circle)
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