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Is there a quick way to turn that triangle into a square by "splitting" the selected vertex? Something like an extrusion that will follow the edge.

the red arrow is where the new vertex should go, the green one is the position of the edge that is linked to the new vertex The red arrow is where the new vertex should go, the green one is the position of the edge that is linked to the new vertex.

I know I can make something similar with the hotkey V to separate the vertex but i still have to fill the gaps:

enter image description here

I also know that I can simply use the knife tool to add an edge and then remove the edge between the two triangles to have a quad.

The final result I want to achieve:

the final result I want to achieve

It is simply that I wonder if there is a way to slide a vertex with some of its edges (not all) or if it is not possible.

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Yes, you can just hit Alt + v to split it while hovering over the edge you would slide it on and then click g to slide it:

enter image description here

It's also in Mesh -> Vertices menu(Ctrl+v) as Rip Fill

enter image description here

Edit: I just realized this is not completely the desired result you show in the picture, because there is still an edge left where it was originally and you would need to dissolve it with Alt+x, however I think this is the closest thing. I think it would not make sense to have a tool that would remove the edge, because in that case it would in most cases leave untidy geometry, that might be best to avoid, so I can understand why there is no such a tool precisely as you described.

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