Once you familiarize yourself with the basic axioms and theorems of geometry, you will be able to see how they apply to the proof of any particular problem that you may be working on.
A logical chain of steps, supported by postulates,defentions, and theroems, to prove a statement is true. -ERA -2-
Indirect proofs are a very useful tool, not just in geometry, but in many other areas - making it possible to prove things that would be hard or impossible to prove otherwise. An example outside of geometry is the fairly simple proof, often found in high school algebra textbooks, that the square root of 2 is not a rational number.
A direct proof of the infinity of primes would require what is essentially a formula to calculate the Nth prime number; such a formula isn't even guaranteed to exist. It's possible to formulate a proof of the infinity of primes that would be, in a sense, direct. A direct proof that the square root of 2 is irrational is impossible, because the irrational numbers aren't defined in any direct way - just as the real numbers which aren't rational. So to prove that the square root of 2 is irrational, we have to prove that it's not rational, which requires indirect techniques.
Prove (verb). A prosecutor has to prove the defendant committed a crime. He presents the proof to the jury in order to prove his case.Another, job-specific verb form of proof is in my industry, journalism, where we will say "Would you proof this page?" In this case proof is a shortened version of the verb proofread. This probably is not in Webster's.
Riders, lemmas, theorems.
Theorem 3.9. If two lines are perpendicular, then they intersect to form 4 right angles. You would do a proof by using your hands.
A criminal conviction in court would normally require proof.
A deed would prove ownership.
postulates cannot be proved, they are the base of geometry and there isn't anything to prove it with. if the postulates were wrong then all of euclidian geometry would be wrong. that is like saying how do we know the English language is correct, it is the basis for communication and if it wasn't, then how would speaking the language work?
You may optionally register a copyright (in some countries) as documented proof of ownership, but you generally have no obligation to prove you own it. If a defendant wants to claim you do NOT own it, they would have the burden of proof.
This is a "proof by contradiction", where the evidence would fail to support the reverse assumption, giving credence to the original hypothesis.
A person would have to submit their coin to an authentication service. Then the service would prove if the coin is real or not and give a proof of authenticity if the coin was authentic.