Is this proportionality proof erroneous?

Solution 1:

Revised and updated answer

I did not examine carefully your assertion that ${a}^2$ is directly proportional to $bc$

Actually, a is jointly proportional to $b$ and $c$ which means

$a \propto bc$

Here are two examples:

Distance travelled varies directly with speed for a given time.
Distance travelled varies directly with time for a given speed.
Distance travelled varies jointly with speed and time, distance $\propto$ speed$\times time$

To take another example with both direct variation, and inverse variation,

Time taken for a journey varies directly with the distance for a given time
Time taken for a journey varies inversely as the speed, fpr a given time
Time $\propto$ distance/speed

[ It just happens that in the above two examples, the constant of variation is $1$ ]

Added: Proof for the original example

Let $a$ change to $a_1$ while $c$ is unchanged, and $b$ changes to $b'$, then $a/a_1 = b/b'$

Then let $a_1$ change to $a_2$ while $b'$ is unchanged, and $c$ changes to $c'$, then $a_1/a_2 = $c/c'$

Multiplying the two, $\frac{a}{a_2} = \frac{bc}{b'c'}$
so $a\propto bc$