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More posts by @Eichhorn212

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@Margaret771

Obviously you want exact snaps and absolutely no angle measurements.

Here's one patience exercise:




Your two crossing lines are blue and red, the circle is green. Start by drawing a horizontal (drag+hold Shift) line from the crossing point and duplicating the blue line. The duplicate is later seen as cyan. Be sure you have smart guides and snap to points in use. No other snaps!
Select blue and red lines. With the rotation tool rotate the blue and red line around their crossing point; make the blue line horizontal this way. Leave the duplicate intact (drop out of the selection).Drag the cirle into the angle so that the bottom anchor sits on the blue line.
Make a 90 degrees rotated copy of the red line. Drag its end to the center of the circle (=deselect all, select red line, drag with white arrow the endpoint of the line). Bring the circle to front and add a new anchor to the circle at crossing A. Remove the rotated copy of the red line.
Make a copy of the red line. Drag it starting from the inserted anchor point (A). Send the blue line to back and insert an anchor to the red copy at crossing B.
Select the circle and the red copy, at first click the circle. Drag with the white arrow the inserted anchor (red copy and circle follow) to the crossing of blue and red lines. Remove the red copy.
With the rotation tool rotate back the circle and red & blue lines. Remove the extra lines.


Done! But quite ineffective when compared to CAD-like extensions which are presented in another answer. Also error prone; dragging an anchor point with the white arrow is needed for snapping and it very easily breaks the shapes. All must be at first deselected, then select with the black arrow the shapes you want to move and with the white arrow drag just an anchor point.

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@Eichhorn212

For a right (90°) angle:

Using Smart Guides (Ctrl/Cmd + U to toggle on/off)



Note. Make sure to turn off Snap to Point, Snap to Grid and Align Objects to Pixel Grid (so that it doesn't interfere with the Smart Guides)

You would also go to Edit→Preferences→Smart Guides and manually add in the angle of your lines, if they aren't already a common angle (90:180)



For an arbitrary angle:

Use this (free) tool/plugin for Illustrator by Astute Graphics called SubScribe which will accomplish this for other arbitrary angles (non 90° angles)

You use the Tangent Circle option



Select the point where the lines intersect and then drop the anchor based on how large you'd like the circle.

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