📂Lemmas

Gabi's Proof of Li

Theorem

If $bdf(b+d)\neq 0$ then $$\frac { a }{ b }=\frac { c }{ d }=\frac { e }{ f } \implies \frac { a+c }{ b+d }=\frac { e }{ f }$$

Description

“Gabi” is nothing else but a word made from two Hanja characters: add 加 and compare 比. Here, the compare 比 is the same as the ‘ratio’ in ratios, making it a theorem where everything is encapsulated in the name.

Proof

$$\frac { a }{ b }=\frac { c }{ d }=\frac { e }{ f }$$

Therefore, $\frac { a }{ b }=\frac { e }{ f }$ and $\frac { c }{ d }=\frac { e }{ f }$. If we multiply both sides of $\frac { a }{ b }=\frac { e }{ f }$ by $bf$,

$$\frac { c }{ d }=\frac { e }{ f }$$

and, if we multiply both sides of $\frac { c }{ d }=\frac { e }{ f }$ by $df$,

$$cf=de$$

Adding up the two obtained equations on both sides gives us

$$(a+c)f=(b+d)e$$

Dividing both sides by $(b+d)f$ results in

$$\frac { a+c }{ b+d }=\frac { e }{ f }$$

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