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When constructing a logo, I use the Divide tool a lot, in order to cut shapes. In the following image, I am trying to cut the curve with a straight line.

enter image description here

After using the Divide tool on these two objects, the result looks like this:

enter image description here

Of course, now I want to cut the point off, so I use the Direct-select tool and I click on the anchor points and I press delete.

enter image description here

So, this is the basic shape I am looking for. However, if you can see, it fills in extra areas above the shape. Once I delete that extra area, it deletes part of the shape that I want, creating a mess entirely. How do I work around this problem?

enter image description here

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You've missed some of the anchor points in the shape that you are trying to remove. An unclosed shape can still have a fill which is why you are seeing this result.

To avoid this issue you can ungroup your shapes after performing the divide so that you don't need to use the direct selection tool, you can just select each shape as a whole using the 'normal' selection. Another quick workaround (if you need or want to use direct selection) is to click on any one point or line in the shape and hit delete twice - this will remove the whole shape.

As a side note; I would say that using divide as your 'go to' pathfinder function is not usually best practice. You can get cleaner, quicker results by getting familiar with the rest of the pathfinder palette and then using the best one for the job in each case. Divide can be a little messy, in my opinion.

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