Proof by contradiction

In a standard classical argument for P, not-P is added to the accepted premises. Incompatible consequences then rule out that added assumption and establish the conclusion.

The incompatibility must follow through valid steps from those assumptions. Using the desired conclusion along the way, silently changing conditions or confusing different objects does not complete a contradiction proof.

References: [1]

I.6 of the Elements establishes that equal angles in a triangle have equal opposite sides. Euclid supposes the sides unequal, cuts a matching length from the longer side and joins the new point.

The earlier side-angle-side result would make a smaller internal triangle equal to the original, contradicting the whole being greater than its part. The diagram aids understanding, but the cutting, correspondence and earlier result supply the argument.

References: [1]

IX.20 begins from any assigned collection of primes and adds one to a common multiple. If that number is prime it supplies another prime; otherwise the argument examines one of its prime divisors.

That divisor cannot belong to the original collection, since it would then divide both the multiple and a number one greater, hence divide one. Construction and exclusion work together; the argument does not claim that a product of primes plus one must always be prime.

References: [2]