Project vertices on a circle

Hi there, i have a problem here which gave me headaches for a long time, i want to project only these vertices ( not the whole model ) straight forward onto the faces of the circle, i tried all the snapping option combinations and did not succeed, i will attach a picture as well as the blend file.
I tried the shrinkwrap modifier too, but it does not give me a straight projection,
At 17:00 on this video the face snapping works well, i tried following this too but did not succeed

can anyone help ? thank you !!

blend source file.

Make a new circle as a separate object, and then apply a Shrinkwrap modifier to it, and set the target to the large cylinder. In the Shrinkwrap settings, set the mode to project. That should work for what I think you want…

I tried this but it does not work because my cilinder is not aligned with the big circle in one of the x or y axis, and the projection will not be straight forward.

Hi, try this:

Nice, the only problem is that i must join those 2 objects, this video shows how to extrude a cilinder from another one, but if i find out how to join these 2 meshes i will be able to project other meshes too on a surface.

In the tutorial he mentions that:“this will only work in shaded view”
and in 17;21 he show something needs to be activated.

Blender 2.8 is a bit different from the versions of the video, but i tried every combination of options, also enabled that feature which you see at 17:21 ( project individual elements ).
I am wondering, what does he mean by shaded view ? isnt that the normal solid view ?

Then as was all ready been mentioned Shrink wrap should work:

Nice, this is the issue i encounter when i try the shrinkwrap, see image attached, i have also attached the source file so you can try it yourself, i want to project only the selected vertices, not the whole model, and it get’s projected on an angle( not straight )

source file

Hi, i have set the origin to object and lined the two object, the small cylinder i moved slightly in to the large one, there is slight taper but better fit.
here is the file:link

Thank you !!