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I have two nodes that create a curved edge. I want to add a new node where the handles for these nodes would intersect if they were extended, and then delete the original two nodes. This would turn the curve into a sharp edge.

enter image description here

I can add a new node, eyeball it into place, and delete the two original nodes, but is slow, and seems inexact. Is there a better way?

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It is not possible to use Live Corners after selecting one of the points?

Here the explanation from Adobe.com

live corners

If not,

  • Duplicating the figure a couple of times, following each line as a displacement guide
  • After selecting both duplicates, with the Shape Builder Tool and pressing Alt delete the surplus
  • Unite this shape with the original one from Pathfinder

deplacement

  • I see the Live Corners handle on some corners, but not this specific corner. – Scribblemacher Jul 13 '18 at 12:26
  • This happen when you edit one of the points – user120647 Jul 13 '18 at 12:29
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See image 1. Circles A,B,C and D contain the anchor points of the blue shape. Let's assume you want to continue straight line segments AB and DC to form a sharp corner.

enter image description here

  1. Draw lines AB and DC. To do it exactly have smart guides and snap to points ON, no other snaps!

  2. Holding Shift at the same time drag the new line objects with the normal selection tool to bigger sizes to make a crossing

enter image description here

  1. With the anchor removal tool remove anchor B and with the anchor type conversion tool make C a corner point

  2. Drag C to the crossing of the lines with the direct selection tool

NOTE: This receipe is useless, if you literally want to find the point where the extensions of the handles meet, like in this:

enter image description here

Blue shape can be curved in both sides of B and C, but have tangents, at least one sided tangents. Finding those tangents need something smarter - for example a third party plugin. Check, if Astute Graphics has a good one. I haven't one.

Alternatively you can goto Inkscape. There's a path visualization extension "Draw handles" which really inserts to the image all handles as lines. It's no problem to extend them:

enter image description here

Then you can draw the exact needed add-on shape and make an union:

enter image description here

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