Being Equal To Something Greater Than Yourself: Another Fallacy

In our previous post , we saw a mathematical fallacy and hopefully you enjoyed it. In this post we're going to see another fallacy. This one's one of my favorites because it doesn't rely on an algebraic technicality like division by zero. What makes this one work is far more devious!

The title of this post is quite thought-provoking [hopefully]. Can you be equal to something that is greater than you? No. That would be illogical. But we're going to use a loop-hole to 'prove' that. Brace yourself!

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$ABCD$ is a rectangle and $CE=CD$. $FJ$ and $FI$ are the perpendicular bisectors of $BC$ and $AE$ respectively. And $J$ and $I$ are the midpoints of $BC$ and $AE$ respectively. We're going to join $F$ with $A$, $B$, $C$ and $E$ and we're ready for 'proving'!

Now, since $F$ is a point on the perpendicular bisector of $AE$,

$FA=FE$

Because of a similar argument,

$FB=FC$

And, $AB=DC$ because they're the opposite sides of a rectangle.

Notice that we've just proved that $\triangle ABF$ and $\triangle ECF$ are congruent [$\text{SSS}$ congruence].

So, $\angle ABF=\angle ECF \cdots(1)$

And also $\angle JBF=\angle JCF \cdots(2)$ because $FC=FB$.

Now subtract $(2)$ from $(1)$.

We have $\angle ABF-\angle JBF=\angle ECF-\angle JCF$

In other words, $\angle ABJ=\angle ECJ$.

But that's impossible! Because $\angle ABJ$ is a right angle and $\angle ECJ$ is an obtuse angle! $\angle ABJ$ is equal to something greater than itself. Can you spot the fallacious step?

Now's the time I tell you stop reading and think about this for a while [even though most of you don't listen :)]. So, start thinking!

Congratulations to everyone who gave this problem some time and thought! Before we find out our mistake, I want to ask you a question.

Is it necessary or compulsory to draw an accurate figure in order to do a geometric proof?

No. Actually you can't ever draw an accurate figure or diagram to be completely strict. We draw figures in our proofs as references. Figures help us follow the arguments presented in a proof. But they are not a necessity.

The figure I gave you in the beginning of the post is inaccurate. It is misleadingly inaccurate!

So, did I commit a crime?

No! We just said that it isn't compulsory to have accurate figures. It isn't wrong to draw inaccurate figures in proofs. But it is wrong to draw an inaccurate figure and make incorrect implicit assumptions from it. And that's what we did.

Everything up until the last step of this 'proof' is $100$% correct [if I haven't made any typos]. But in the last step we assumed something without proof and that led to this absurd conclusion.

When you draw the an accurate enough figure, you'll see that even though $\angle ECJ$ is an obtuse angle, it is not equal to $\angle ECF-\angle JCF$. We only assumed this from the figure [which was inaccurate]. We didn't prove it. And look what it did!

Here is a more accurate depiction of the whole situation:

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Now you can see that $\angle ECF-\angle JCF\neq\angle ECJ$. Our faith in reality is restored!

If you want to check if everything else is okay, you can apply the previous arguments in the new picture. You'll see that everything works. The sides really are equal and the triangles are congruent as well. Everything works up until the last step where we assumed something unwarrantedly.

Unwarranted assumptions are really dangerous. That's why you sometimes see some kind of warning against them in some Brilliant problems [Details and assumptions: Be very careful about how you draw the picture. Don't make any unwarranted assumptions].

Can you resolve this fallacy without drawing an accurate figure? This is a good exercise for people who solve geometry problems without drawing figures [personally, I can't solve geometry problems without drawing figures]. Feel free to post your comments in the comments section below.