Little Joseph once said:
I was once asked to prove that (PROOF ONE) in a right triangle, the sum of the squares of two sides equals the square of the third. Smart little rascal that I am, I proved it with ease. I was then asked to prove that (PROOF 2) if the sum of the squares of two sides of a triangle equals the square of the third side, then that triangle is right. What a stupid question! I'd already proved it once. Why ask me to prove it again? I just wrote down "Already proven."
Comment on Joseph's second proof. Is he as clever as he thinks he is?
Elementary Logic :)
hypothesis - if a cow, then a mammal
if true then
converse - if a mammal then a cow
Not sure
( It is the converse we have here :)
inverse - if not a cow, then not a mammal
Not sure
contrapositive - if not a mammal, then not a cow
True.
aah ok
Joseph's second proof is not a valid proof. Simply writing "Already proven" is not a convincing or valid argument. In mathematics, each statement or theorem needs to be proven individually and cannot be assumed to be true just because it has been proven in a different context.
To properly prove that if the sum of the squares of two sides of a triangle equals the square of the third side, then that triangle is right, Joseph would need to provide a separate proof for this statement. He cannot rely on the previous proof he provided for the first statement.
A valid proof would involve using the properties of triangles and the Pythagorean theorem to show the relationship between the sides of the triangle and how it implies the triangle is right.