Theorems, Corollaries, Lemmas
What are all those things? They sound so impressive!
Well, they are basically just facts: some result that has been arrived at.
- A Theorem is a major result
- A Corollary is a theorem that follows on from another theorem
- A Lemma is a small results (less important than a theorem)
Here is an example from Geometry:
Example: A Theorem, a Corollary to it, and also a Lemma!
An inscribed angle a° is half of the central angle 2a°
Called the Angle at the Center Theorem.
Proof: Join the center O to A.
Triangle ABO is isosceles, so Angle OBA = Angle BAO = b°
And, using Angles of a Triangle add to 180°, Angle AOB = (180 - 2b)°
Triangle ACO is isosceles, so Angle OCA = Angle CAO = c°
And, using Angles of a Triangle add to 180°, Angle AOC = (180 - 2c)°
And, using Angles around a point add to 360°:
|Angle BOC||= 360° - (180 - 2b)° - (180 - 2c)°|
|= 2b° + 2c°|
|= 2(b + c)°|
Replace b + c with a, we get:
Angle BAC = a° and Angle BOC = 2a°
And we have proved the theorem.
(That was a "major" result, so is a Theorem.)
(This is called The Angles Subtended by the Same Arc Theorem, but it’s really just a Corollary of The Angle at the Center Theorem)
Keeping the endpoints fixed ... ... the angle a° is always the same, no matter where it is on the circumference:
So, Angles Subtended by the Same Arc are equal.
(This is sometimes called The Angle in the Semicircle Theorem, but it’s really just a Lemma to The Angle at the Center Theorem)
In the special case where the central angle forms a diameter of the circle:
2a° = 180° , so a° = 90°
So an angle inscribed in a semicircle is always a right angle.
(That was a "small" result, so it is a Lemma.)
Another example for you, related to Pythagoras' Theorem:
If m and n are any two whole numbers and
- a = m2 – n2
- b = 2mn
- c = m2 + n2
then a2 + b2 = c2
|a2 + b2||= (m2 – n2)2 + (2mn)2|
|= m4 – 2m2n2 + n4 + 4m2n2|
|= m4 + 2m2n2 + n4|
|= (m2 + n2)2|
(That was a "major" result.)
a, b and c, as defined above, are a Pythagorean Triple
From the Theorem a2 + b2 = c2,
so a, b and c are a Pythagorean Triple
(That result "followed on" from the previous Theorem.)
If m = 2 and n = 1, then we get the Pythagorean triple 3, 4 and 5
If m = 2 and n = 1, then
- a = 22 – 12 = 4 – 1 = 3
- b = 2 × 2 × 1 = 4
- c = 22 + 12 = 4 + 1 = 5
(That was a "small" result.)