As I understand - you need a new vertex on edge exactly in the place where it intersects with another edge.
Here is it.
Warning: orange is a selection color in this examples.
Make a reference oblect:
![[Image: 00.png]](https://i.ibb.co/PzRBrbG/00.png)
Select edge that willl be cutted:
![[Image: 01.png]](https://i.ibb.co/dtYd5Hq/01.png)
Cut it in half (create new vertex):
![[Image: 02.png]](https://i.ibb.co/dMcZG4H/02.png)
Click RMB & select "Intersect [RMB]"
![[Image: 03.png]](https://i.ibb.co/VTyLxLB/03.png)
Pick edge to set direction of new vertex movement & click RMB
![[Image: 04.png]](https://i.ibb.co/7yQSPKL/04.png)
Pick vertex to move & click RMB
![[Image: 05.png]](https://i.ibb.co/5K2KxgW/05.png)
Pick plane on the reference oblect (to which we will move vertex) & click RMB
![[Image: 06.png]](https://i.ibb.co/Tg9r9wY/06.png)
Pick point on edge that intersects & click RMB
![[Image: 07.png]](https://i.ibb.co/WsP6r5D/07.png)
Done.
Here is it.
Warning: orange is a selection color in this examples.
Make a reference oblect:
![[Image: 00.png]](https://i.ibb.co/PzRBrbG/00.png)
Select edge that willl be cutted:
![[Image: 01.png]](https://i.ibb.co/dtYd5Hq/01.png)
Cut it in half (create new vertex):
![[Image: 02.png]](https://i.ibb.co/dMcZG4H/02.png)
Click RMB & select "Intersect [RMB]"
![[Image: 03.png]](https://i.ibb.co/VTyLxLB/03.png)
Pick edge to set direction of new vertex movement & click RMB
![[Image: 04.png]](https://i.ibb.co/7yQSPKL/04.png)
Pick vertex to move & click RMB
![[Image: 05.png]](https://i.ibb.co/5K2KxgW/05.png)
Pick plane on the reference oblect (to which we will move vertex) & click RMB
![[Image: 06.png]](https://i.ibb.co/Tg9r9wY/06.png)
Pick point on edge that intersects & click RMB
![[Image: 07.png]](https://i.ibb.co/WsP6r5D/07.png)
Done.
![[Image: 08.png]](https://i.ibb.co/hKqWq47/08.png)