# Theorem A. (the Golden ratio and the Key Triangle)

The Golden ratio is produced with the Key Triangle

(a+c) / b = (1+√5) / 2 = φ, with any a, b, c ∈ Z_{+}.

This is the same as: any non-negative integer divisible by 3 can be divided into the legs of a right triangle with ratio 1 : 2, and when comparing the sum of the shorter leg and the hypotenuse against the longer leg, the result is always the golden ratio.

*Image 6. The Key Triangle (a right triangle with legs 1 : 2)*

## Lemma a. (the Hypotenuse of the Key Triangle)

In Key Triangle is valid

c = a √5, with any a, c∈ Z_{+}.

#### Proof (Lemma a.)

Because of the Pythagorean Theorem is valid

c ^{2} = a ^{2} + b ^{2} = a ^{2 }+ (2a) ^{2 }= a ^{2} + 4 ^{2} = 5a ^{2},

therefore

c = √( 5a ^{2}) = a√5.

∎

#### Proof (Theorem A.)

The proof is by mathematical induction.

1) the basis: a = 1

φ = (a + c) / b = (a + a√5) / 2a = (1 + √5) / 2, that is true.

2) the inductive step

a) inductive hypothesis: a = n

φ = (n + n√5) / 2n (= (1 + √5) / 2).

b) a = n + 1

φ = ((n + 1) + (n + 1) √5) / (2 (n + 1)) = (n + 1) / (2 (n + 1)) + ((n + 1) √5) / (2 (n + 1)) = 1 / 2 + (√5) / 2 = (1 + √5) / 2 = (n + n√5) / 2n.

Since both the basis and the inductive step have been performed, by mathematical induction, the statement φ=(1+√5)/2=(a+c)/b holds for all positive integers a, b, c.

∎

It is demonstrated above that the King’s Chamber produces the golden ratio with all possible units of measurement. In fact, the King’s Chamber is the Golden ratio. We will take this result even further: given that “*any non-negative integer divisible by 3 can be divided into the legs of a right triangle with ratio 1 : 2, and when comparing the sum of the shorter leg and the hypotenuse against the longer leg, the result is always the golden ratio*” – could this be the new accurate definition of the golden ratio?^{5}

* *

^{5} Another worthy question is whether the golden ratio could also be defined as an average of square roots 1 and 5.