Are You Inside Or Not?

I'm back with another mathematical fallacy which I found to be dangerously devious when I saw it for the first time. To be honest, I couldn't spot the mistake in this one until I'd drawn an accurate picture. Before we dive into this fallacious proof, a word of advice: never assume anything without proof!

Imgur Imgur

We have a circle with center $A$ and radius $r$ . $C$ is a point inside the circle. $B$ is a point such that $AB\cdot AC=r^2$ . We're going to draw the perpendicular bisector of $BC$ and let that bisector intersect the circle at $E$ and $D$ . $F$ is the midpoint of $BC$ .

Now on to the proof!

$r^2=AB\cdot AC$

$\Rightarrow r^2=(AF+BF)(AF-CF)=(AF+CF)(AF-CF)=AF^2-CF^2 \cdots(1)$

Now we're going to use the Pythagorean theorem.

$AF^2=AE^2-EF^2=r^2-EF^2$ and

$CF^2=CE^2-EF^2$ .

Let's make these substitutions in $(1)$ .

We have,

$r^2=r^2-EF^2-CE^2+EF^2$ .

Now, if you do the calculations, you have :

$CE=0$

Take a moment to understand what this statement is saying. Take a look at the picture if you have to.

I know it's hard to believe but this means that $C$ is actually on the circle. But that is insane. By definition, $C$ is a point inside the circle. What went wrong?

Before you read the rest of the article, think about this for a while. Review the steps. And most importantly, don't assume anything without proof.

Now how do we solve this apparent contradiction? As I said earlier, if you try to draw an accurate picture, you'll see what's happening here. But I find the picture here a bit uninteresting. So, draw an accurate picture if you want to. I'm not going to do that here. Instead, we're going solve this with pure logic. Let's begin!

Okay, since $r>AC$ , we have $(r-AC)^2>0$ .

Rewrite this as $r^2+AC^2>2\cdot r \cdot AC$ .

Multiply both the sides by $\frac{1}{2 \times AC}$ to get:

$\frac{1}{2}(\frac{r^2}{AC} +AC)>r$

Remember that $\frac{r^2}{AC}=AB$ . So, we have:

$\frac{1}{2}(AB+AC)>r$

$\Rightarrow \frac{1}{2} (AC+BC+AC)>r$

$\Rightarrow AC+\frac{1}{2}\times BC>r$

$\Rightarrow AC+CF>r$

$\Rightarrow AF>r$

This actually solves the whole thing! Because it tells us that the point $F$ is actually outside the circle and the points $D$ and $E$ don't even exist! We only assumed they existed but we never proved it. If these points don't exist, all the arguments mentioned above are meaningless.

As we'll see again and again, implicit [& incorrect] assumptions will come back to haunt us.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
list

Note: you must add a full line of space before and after lists for them to show up correctly

paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

Math

Appears as

Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I'm back with another post. Feel free to post your feedback in the comments section.

I see that Cosines Group is "For Level 2-3 members, an immersion in concepts beyond the standard curriculum". So, also let me know if this falls outside the Cosines Group range.

Mursalin Habib - 7 years, 5 months ago

Hi Mursalin, can you suggest me good way to start studying Euclidean geometry?

Snehal Shekatkar - 7 years, 5 months ago

Sure! I had my beginning in geometry from my school textbooks. The geometry curriculum in our country is good. And the books had a lot of problems for us to work on.

For starters, I'd recommend the 'Geometry for Americans' section in Paul Zeitz's book, "The Art & Craft of Problem Solving" (2nd edition). For solving Olympiad level problems, THE book I can recommend you is 'Geometry Revisited' by Coxeter and Greitzer.

The most important thing while studying geometry is making sure that you understand formal proof structures and checking if you can apply the theorems you've learnt. For example, when you have an angle bisector and anything with ratios, you have to start thinking about the angle bisector theorem. And there's no better way to do that than solving problems, lots of them.

I hope this helps!

@Mursalin Habib – Thanks Mursalin.. but I can see that these books are quite expensive.. how did you get them?

@Snehal Shekatkar – Geometry revisited is the second link on the first page of a google search. I'm not sure if it's legal or not. Anyway, I contacted the BdMO [Bangladesh Mathematical Olympiad] and ordered the books.

If you don't want to buy the books, no problem! All you need is the index or the 'contents' page. And you can view that on Amazon. Then you can learn about those topics from free sources like Wikipedia, Mathworld or Aops. Yes, that way you can't learn the topics from the author's own words, but you can learn. There are free resources all over the internet.

Nice one :) Got me trick for a while too :)

Happy Melodies - 7 years, 5 months ago

I got a little suspicious when I saw that $F$ was assumed to be inside the circle. Since as $C$ approaches $A$ , then $B$ approaches infinity; surely $F$ cannot go left of the intersection of $AB$ and the circle...

Turns out I was right :)

Daniel Liu - 7 years, 5 months ago

That is actually a great way to think about it. The more time you spend with fallacies like these, the better you get spotting the fallacious steps. Good job!

I didnt think of it that way...nice!!

Tanya Gupta - 7 years, 3 months ago

So we can't argue with no proof before. Really nice post!

Muh. Amin Widyatama - 7 years, 5 months ago

so coool!!

Jordi Bosch - 7 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$