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!
We have a circle with center and radius . is a point inside the circle. is a point such that . We're going to draw the perpendicular bisector of and let that bisector intersect the circle at and . is the midpoint of .
Now on to the proof!
Now we're going to use the Pythagorean theorem.
Let's make these substitutions in .
Now, if you do the calculations, you have :
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 is actually on the circle. But that is insane. By definition, 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 , we have .
Rewrite this as .
Multiply both the sides by to get:
Remember that . So, we have:
This actually solves the whole thing! Because it tells us that the point is actually outside the circle and the points and 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.