 Now I'm going to talk about straight-up proofs when it comes to geometry and we're specifically dealing with two-dimensional geometry Okay, now this is the problem laid out in point form The first thing they're giving you is B is the midpoint of AC C is the midpoint of BD D is the midpoint of AE CE the length CE is given as q and they want us to prove That AE is equal to 5 q over 3 now when you read this in one shot. It's crazy. It's chaos. There's letters everywhere So what you do the way you want to do the way you approach Solving proofs or doing proofs is you read the question once just to understand You know what's coming at you just to get an idea of what is coming at you then what you do is you go back and you take each point on at a time and Draw the diagram if you want if you need to or think about the problem if you need to So you can break it down properly to deal with whatever you need to prove So let's go to first point This is the first thing I'm going to think about B is the midpoint of AC now I've done this problem once before with just solving for it in problem form But I'm going to do it again going through all the steps to lay out the problem again So B is the midpoint of AC. So what we're going to do is going to draw this We need to line AC and we're going to put B as a midpoint. So we're going to go A and C and B is the midpoint of AC. So we're going to put pop this thing down Right in the middle now. There's our line AC with B is the midpoint now When they see B is them when they say B is the midpoint they mean that B Breaks this line in two equal parts. So this guy equals this guy and what you should do whenever you're doing proofs Or whatever you're doing problems is use symbols to remind yourself what each Part that you've already figured out is so right away visually we can see that this line equals this line We don't have to continuously go back to the problem to try to figure out what equals what So make marks according to what you like the type of marks you like to say what line equals what? Ticks are easy way to do it. If you have two different types of lines use double ticks, okay? so This line equals this line we dealt with this point then we're going to go down to this one C is the midpoint of BD. We have C. We got B. We don't have a D So C is the midpoint of BD. So D's got to be over here and If this is the midpoint of these two guys then this guy equals this guy, which means it equals this guy So right away you got three additional pieces of information, but just going through two lines D is the midpoint of BE. We have a D. We got a B. We don't have an E So E's got to be way over there. So D is the midpoint. D is the midpoint of BE So E's got to be Here. Now if D is the midpoint of BE, now don't worry about if these things aren't proportional What you know is that this guy is the midpoint between this guy and that guy So what you do is you go, okay, this guy is equal to this guy So we've dealt with this guy as well Now this is a piece of information they're giving us. C E is equal to Q. Now where is C E? C E is equal to Q. I'm just going to move this guy up a little bit. So our line doesn't go through it So C E is equal to Q What we've done right now is transferred the whole problem Into visual form so we can deal with the question that they're asking us to prove this thing So right now what you could do is step back and take a look at this stuff and figure out what you need to What you need to find out. AE AE AE is equal to 5 Q over 3. We have to prove this This is what we need to prove so we're going to have to start from one of these pieces of information Okay, now let's take a look. What are we going to start off with? Well, take a look at this This guy is equal to this guy, but we don't know what this guy is equal to so what we can do is From this relationship Figure out what DE is equal to now if this is equal to this then This length is equal to this length And if we break this guy in two Put our own midpoint in there. You can call it w or anything you want Okay, then you know this length must be equal to This guy and that length must be equal to that guy. They're all equal so you can just go take a take So what you can do is say Breaking this down right out your own proofs now what you should be doing when you're breaking this stuff down is Writing down the points now AB AB now this is what you will write down as your solution AB is equal to BC is equal to CD Because of the midpoint property Midpoint property now these are the types of types of Things that you're going to get Marks for when you do the proofs. Okay, so that's your first point. This is your actual solution Your actual proof that this guy is equal to this guy is equal to this guy because of the midpoint properties that we figured out in here What we have here we can say dw you can introduce your own variables if you like dw is equal to w e so d w is equal to w e Which is equal to all of those guys above AB BC and CD and again this relationship is the midpoint property Whenever you do proofs the way it works is you make a statement and you give the reason why the statement is valid So that's your statement and these are our reasons why the statement is valid. So right now. We've got all these guys well All we have to do is figure out what each one of these equals to because once we do that All we do is just multiply by five of these because there's five of them one two three four five So how we're going to figure out what one of these guys is in a form that's in the question In the proof now in the proof they're using q so we have to go back to q now We take a look at q we say okay How we're going to use q to figure out what each one of these is well the links from here to here is q and This this link is broken into three different segments. So each one of these segments must be q divided by three So what you can do is say? CD is equal to d w is equal to W e Which equals q over 3? Okay What you can say for this is this is given given and the midpoint property Because you use the information that they gave you and you use the information that you fit that the midpoint property Relationship you use to figure out what each one of these segment was or what they equal to each other Okay, so now you have what each one of these is and Since each one of these is the same then the links from a e the links a e Must be equal to one two three four five of these things Q over three multiple got together. So a e is equal to five times q over three Therefore a e is Equal to five q over three and that's your proof That's that's the simplest proof you can go through It's basically the way you should think about proofs is it's you laying down the thought processes that you're going through Trying to figure out what something is okay So it's basically laying out what your thoughts are and that's the power of proofs because it allows you to Quantify your thoughts in an orderly fashion that anyone coming up here Would be able to follow and come up with the same answer so they would understand it Now this is two-dimensional proofs. I'm going to go ahead and go into actually this is one-dimensional proof Sorry, because this is in one dimension. It's just on a straight line. What we're going to do We're going to go into a two-dimensional proof Which is the x-axis and the y-axis and deal with that Okay, and that's going to be coordinate geometry this guy had no coordinates involved with it