This Proof is Lying to You

[00:00:00]Okay, so we're going to look at a really interesting deceptive geometric proof.

[00:00:04]Now, I like this because every step seems perfectly logical along the way and yet the conclusion we're going to reach is certainly false. I don't think it's at all obvious where the mistake is. So, our setup is we start with a square then we're going to draw in a line segment on the outside like this so that this angle isn't zero and so this has the same length as the side length of our square. Then we're going to connect this point here to the top left of our square. So, we connect to the top left corner of our square like this.

[00:00:32]Now, we're going to draw in some perpendicular bisectors. So, we start with the perpendicular bisector of this top side of our square here. So, we go through the middle perpendicular and this is going to bisect the bottom side as well and it continues down here. Then we're also going to draw in the perpendicular bisector of this new line that we've drawn as well. So, just label this make that clear that's a right angle and then the perpendicular bisector is going to start somewhere over to the right here. If we do this perhaps in a dotted line, you can see the two of them are going to meet somewhere just outside of the square down here like this and we'll again draw this in just to make it clear that that's a right angle. So, now we've got some more sides. We're going to draw in a couple more so we can draw from here to here and also from this corner down to this point at the bottom here. So, then we're going to form lots of different similar triangles that we'll work with in a moment and then from here all we need to do is draw in a couple more sides. So, from this point at the bottom up to our new point that we formed at the top there, we're going to red now for this.

[00:01:37]And we'll also draw in a diagonal from this top left point down to our new point at the bottom there.

[00:01:43]And what we're going to be interested in now is this triangle on the left. If I just go dotted red on the outside this side length, short one, and this side of the square here.

[00:01:54]You can You this forms a triangle on the left and then we're going to claim that this triangle is actually going to be similar to this triangle on the right-hand side. So, if we just draw say that we've got this side, which is the one that we drew in at the very start.

[00:02:08]We've got this short one here, and then we've also got this diagonal that we drew earlier.

[00:02:12]So, you can see first of all that we've got this side length here is certainly because just by construction at the start, we chose this to be the same length as the side length of the square.

[00:02:21]So, it's going to be the same as this length here cuz this is also one of the sides of the square. And then we can look at perhaps these long diagonals next. So, if I just draw out a copy of this triangle where we've got the dotted blue perpendicular bisector there. If we draw out a smaller copy of this just so we can label it a bit more clearly. So, we've got the perpendicular bisector there. Then we've got these two diagonals following here.

[00:02:46]So, we've got cuz it's the perpendicular bisector, we know that both of these angles are equal. They're both 90°. We also know that this side length is going to be equal to this side length here.

[00:02:56]And we've also got this side in the middle is shared between these two right-angle triangles. So, we've got two similar right-angle triangles. So, then we can label let's say with the two lines in red now, these two lengths are going to be equal. So, we're saying that these two red diagonals are going to be equal to each other. And then we'll do something very similar to show that these two remaining shorter sides in our red triangles are also going to have the same length. So, if we draw out a copy of this triangle at the bottom, then we've got these two sides at the bottom of the red dotted short sides. And can you see here we've got the perpendicular bisector.

[00:03:35]This was bisecting one of the sides of the square. So, it's going to also bisect the other side of the square through on the other side. So, we've got now and it's also going to be perpendicular as well cuz this is a right angle. So, that means that we've also got a right angle down here on the opposite side of the square. So, we've again got some right angle triangles. We know that this is going to be a perpendicular bisector. We've got this side here is shared between these two small triangles, which means that these two then are going to be equal to each other as well. So, we can do this for both of these triangles. You could say this is by congruence that we've got two sides and an angle. Or we could even do this calculating the side lengths using Pythagoras to say that the hypotenuse has to be equal in each case. So, then we've got two different similar triangles and all three of these side lengths are the same. So, these two triangles that you congruent and it doesn't seem to have any issues so far in terms of the lengths. But then when we turn to the angles, you'll see that this big angle here has then got to be equal to this big angle here.

[00:04:40]And now this angle's made up of If I just shade this in red, we've got the small angle down here and the small angle down here. So, you can see these two angles appear in our two congruent triangles that are the same here and here. So, we know that these two shaded red angles are definitely the same as each other. And we've got this angle at the top here is actually just a 90° angle. So, if I draw this just a bit bigger, this is certainly 90° and we've got 90° here as well. So, we've got this red angle plus 90° is supposed to be equal to this red angle plus 90° plus there was this other angle that we added at the very start. We perhaps do this one in black. So, this angle that we added at the start and we imposed that this had to be not zero cuz if it was zero, we'd just be working with the side of the square. So, this angle plus 90° plus the red angle has to equal to zero plus 90° plus the red angle. So, you can see we've reached a contradiction that this is impossible that these two angles, which are clearly not equal to each other, seem to be equal to each other. And if we set this angle equal to, let's say, 1°, then we'd actually have a proof that 0 = 1 by considering the angles there. So, well, we'll try and draw this again so that we can understand what's going on and where the mistake is in a moment. But, do if you're interested in solving this, this is a good time to pause the video cuz once you've seen the solution, it's one that you can't unsee.

[00:06:03]So, if we try and draw this as a rough sketch, it's possible we can get a diagram looking quite similar to the one we had before. But, if we take a bit of effort to be a bit more precise, you see we get a subtly different looking diagram. So, we've gone as far as drawing in this line with the same length as the side of the square and also these two perpendicular bisectors. And I think this is actually the first time I've ever used a ruler when creating a video.

[00:06:25]So, that's quite a milestone. So, then if we draw in the remaining side lengths here, we've got the You can see at the bottom these triangles at the bottom, which were quite squashed and quite small, are now much longer and much more spread out because the point where the two perpendicular bisectors meet is actually way further down compared to how we had it on the rough sketch before. And then when we get on to actually drawing in the diagonals to form these two congruent triangles, you see this one on the left looks quite similar to what we had before. There's no particular difference there to note.

[00:06:57]But, the one on the right, you'll see is actually going from this point, you can see it's on the outside of our diagram, whereas earlier it was actually on the inside. So, it was going over to the left of this bottom right corner of the square on this side, whereas now you can see it's on the outside of the diagram if we draw this a little bit more accurately. So, then if we were to draw in the remaining sides of our two similar triangles that have all the same lengths, we've got this length and this length are certainly still equal to each other. The two diagonals are still equal to each other. So, we've still got this perpendicular bisector and these two congruent triangles. And then we've also got these two here for the same reason.

[00:07:34]We've got the perpendicular bisector of the side of the square is still giving us two congruent triangles. So, now we've got instead of the two red triangles being congruent to each other before, we've got the two red triangles just stretched out a bit. And when we get onto the angles, you see we've got the red angle we had before is actually quite a bit smaller than we had before.

[00:07:55]And similarly, the red angle here is These two are still equal to each other cuz we've got congruent triangles at the bottom. And we've still got this 90° angle here as well. And it's true that if we were to add in this extra angle, let's shape this in black. So, if we were to add in this extra angle here, we do indeed have an angle which is bigger than the angle in our triangle. But, because the triangle's angle here is actually the same as the angle on the outside here, we're no longer claiming that this 90° plus the red angle is equal to this 90° plus the red angle plus the black angle. So, there's no contradiction now because the angle that's actually supposed to be equal was on the outside. It was just drawn in the wrong place. So, I think this really emphasizes the importance of drawing a good quality sketch sometimes for geometry. That it's really, really subtle unless you start to draw it a bit more precisely where You can see obviously this diagram isn't also It's not very precise either, but it's a little bit better than the one we had before. And that makes enough of a difference that we no longer get this contradictory result. We've got this angle here is equal to this angle here.

[00:09:01]And it's still equal to this angle here as well. And then this angle that was bigger, we're just no longer claiming that this is equal to this one. So, this contradictory 90° plus red angle plus the black angle is equal to just 90° plus the red angle. We no longer have this contradictory result. All because the angle was just drawn in the wrong place.