Teaching proof in geometry allows students to learn the process of reduction: a process of “reverse-engineering” an induction. Starting with a geometric theorem and tracing the train of thought back through the theorems that the first is logically dependent on, and back to axioms, definitions, and figures, teaches students how to start with a statement and trace it back, in a logically connected train of ideas, to the evidence of the senses.
For professional educators to remove or de-emphasize proof from geometry is intellectually criminal. Besides reduction, there is too much one learns from geometry about logic and reasoning to remove it from the curriculum. But rare is the teacher who is even aware that such a thing as reduction even exists, never mind is aware of how to train students in reduction. There is a decreasing number and proportion who know what deduction and logic are, and how to teach them!
Teaching reduction — and other aspects and issues of logic — does not require some kind of mental contortions, preposterous connections, or abuse of the geometry curriculum. It requires only that one teach geometry correctly (math, like all subjects, should be taught, not in a vacuum, but as a means of teaching reasoning) and that one identify connections to logic and all knowledge when they come up. Opportunities to make such connections come up naturally and frequently.