Did you have a go at drawing any of those concurrent lines or collinear points? Either with instruments, or with software (e.g. Geogebra). If the constructions always ‘work’ are the proofs necessary? What do they do for us? One difference with the software is being able to move the vertices of the triangle and ‘check’ the concurrency – like doing many experiments of drawing, or is it?
Could there be mistakes in the proofs? (I mean the intended proofs – not just any of ‘my’ mistakes!) Related to the question about ‘accuracy’ of Geogebra, it seems to me the software ‘knows’ more than expected. I could not fool it by drawing a third perpendicular bisector into making them truly ‘concurrent’ – only when it was using its own version of perpendicular bisector would it make the intersection one point (rather than three).