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Given a point and a line at an angle, I want to align a third object in between them so they both touch the object without overlap.

Here is an image for reference: enter image description here

How do I align the circle in a way that touches both the star edge and the hexagon side without an overlap. I want the circle to be constrained by those two elements.

The goal is to create the path represented in yellow, so then I can use the Shape Builder Tool. If the alignment is not perfect, it's short, the Shape Builder Tool will not recognise the hexagon being divided. If the circle is overlapping then the cut is not perfect.

A more simplistic implementation could be this: enter image description here

Align the circle so it touches both lines.

This looks something very fundamental to know for logo creation, but I cannot figure it out.

Thanks.

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The solution lies at the intersection of the bisector of the line continuation and the point normal. To do this you must know how to do numeric input in Illustrator works and exploit how illustrator stores previous tasks in numeric input for repetition.

enter image description here

Image 1: Solution is at the intersection of normal and bisector

Finding the bisector of two lines can be done as follows:

  1. Draw a line out from intersection of two lines along one edge.
  2. Double click on line tool (or alt click on screen) to get the numeric input of previously drawn line. Copy the Angle parameter to clipboard.
  3. Draw the second outbound edge
  4. Alt click on intersection with line tool active type + after angle input and paste the stored angle then hit tab to change focus of input. This causes the tool to calculate new angle. type /2 after angle, tab out and hit ok.

Normal of a line is easier just rotate a copy of a line by 90 degrees. Circle can be made by drawing a line from center to edge, opening the line dialog copy the line length. Than alt click on intended centerpoint for numeric input and paste the lengh and *2 on both the dialog inputs and your done.

There are a few caveats with numeric input in Illustrator.

  1. Numeric input can not have more than 2 digits of accuracy
  2. Circles made out of 4 bezier points are only approximately circular

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