Hi all, my name is Ravi Prakash and welcome to the second class of Geometry. Okay. So from here the real geometry will start right. We will do each and every concept of triangles and circles.
Because mostly the questions in this exam like cat and zat are based on triangles and circles right. So we will discuss each and every concept here. Okay.
So let's start. So mostly we will do triangles at first and then we will do circles. Okay.
So let's start with triangles. See in triangles I will first start with all the. All the. Points containing geometric centers.
Okay, so all the points containing the geometric centers Which which point should come under the geometric centers, right? Very important. Okay, please pay attention geometric centers will discuss Okay.
Now there are basically four geometric centers in triangles right first one is first one is Centroid Centroid. Okay. Notice centroid here. Centroid is basically intersection point of intersection point of all the medians.
Okay. So it is intersection point of all the medians. Right. Now what are medians? What are medians?
So again see, you'll get many, many terms like this. Maybe you're not familiar with. You have Fogden geometry or like that.
Right. So you'll get many, many terms like this. So, Medians, right? What are Medians now?
So, Medians contain a word Mid. Mid means joining, line joining midpoint, okay? So, you can remember like this, okay?
So, Medians contain word Mid. Mid means joining midpoint, right? So, what are Medians basically, see?
You take a, take up a triangle like this, triangle AB. So, in triangle AB, now let's say VC midpoint is D, okay? So, D is the midpoint of VC.
That means B and D will be equal. Okay. If you join A here, this A is called the median. Right. This A is called the median.
So here A is the median. Fine. So similarly, I can draw three medians in triangle.
One is A. Another is line joining midpoint of A and C. So let's say B.
Okay. E is the midpoint of A and C. That means A and E are equal.
Okay. So, I can join B and then I can join another let's say F is the midpoint of AB. So, I can join C as well.
Right. So, they will pass through a common point. This is by construction, right. For sure they will pass through a common point.
This point is called syndrome. So, all three will pass through a common point. and this point is called what? This point is called Centroid. Centroid.
So, make this image in mind. A triangle with all the lines joining the midpoints and the intersection point is Centroid. Now, what are the properties of Centroid?
So, properties of Centroid are Centroid divides median Centroid divides the median in the ratio 2 is to 1 okay 2 is to 1 where 2 is the part from vertex right 2 is the part from vertex so to this basically if it is a line and if this is a centroid okay this will divide this line in the ratio 2 is to 1 that means this this part is 2 and this part is 1 so ratio is 2 is to 1 okay Let's name this point as, let's say m. So, let's say m is a centroid. Okay.
That means basically, am is 2, md, the ratio is what? 2 is 2. Okay. So, ratio is what?
2 is 2. Fine. Similarly, for line B, so this vertex part is always 2, right. This m is a centroid here.
This vertex part is 2. So, this part is 2 and this part is 1. So, ratio is 2 is 2, 1. Fine. So, that vertex A, B and C are the vertex of the triangles. a, b and c are the vertex of the triangles. Okay. Centroid divides median in the ratio 2 is to 1. Now, next point.
When you draw a centroid, right? So, you can see here in this triangle right now, there are 6 triangles. 1, 2, 3, 4, 5 and 6, right?
So, triangle is divided into 6 parts. Right? So, that means I can say in when I draw all the medians, so triangle is divided into six triangles of equal area. Right? So right now you can take it as a property of centroid at this point.
Later on when you do the ratio method of triangles area division, right, then you can easily prove this right. So why it is so. Okay.
So right now you can take it as a property of centroid that when this is centroid, centroid comes by joining all these three medians. So triangle is divided into six triangles of equal area. Okay.
And when we do a ratio method of area division, you can easily prove this point, right? Why it is of six triangles of equal area. Fine. So you can remember these three points here. Okay.
In centroid. Okay. Let's move to next point.
Next geometric center. So. Next geometric center will be ortho center.
Okay. So, next geometric center is ortho center. Ortho center.
Now, what is ortho center and all its properties will do? Okay. So, ortho center is basically intersection point of all the altitudes of the triangle.
Okay. This is what? It is the intersection point of all the altitudes.
Altitudes are also called. heights of the triangle right so all the heights of the triangle fine so also center is intersection point of all the heights of the triangle right now let's draw it let's draw it see if this is a triangle a b c okay and i'm drawing a perpendicular right this is the height so let's say ad is the height similarly I can draw perpendicular on AC also. So here B will be the height let's say. B is the height and again let's say C is the height if you draw perpendicular on AB. So let's say C is the height right.
All these are perpendicular. This is a sign of 90 degree fine. So now this is the intersection point right.
Again this is by construction all the three points lines has to be uh have to meet at the common point right. This is called the orthocenter right. Let's name this point as O and this point is called the ortho center of this triangle. Right. So again you can remember like now ortho.
Ortho is a root word right in English. So ortho means what? Ortho means straight right. So which line will be straight? Perpendicular line will be straight right.
So ortho means straight in verbal. See we are studying these are root words right. So it's been straight line okay. So this is a straight line, A is a straight line, B is a straight line, right?
It doesn't matter like, triangle can be like this also. This is a triangle now, AB. So here A is the height.
So we draw a straight line, A is the height, right? And what is the median? This is a midpoint. Okay, so let's say B midpoint is E.
So now A is the median and A is the height, right? So they'll not be same. Okay, they'll be same in one special case of isosceles triangle and one and an equilateral triangle, right? So, we'll study that point later on, okay?
So, that means here, this is an orzo center. What is an orzo center? Intersection point of all the altitudes or height of the triangle, right?
Now, a very important property of an orzo center lies here, right? A very important property of an orzo center. See, the property is, if there is a triangle AB, okay? triangle AB and let's say O is the orthocenter. O is the orthocenter.
Fine. O is the orthocenter. Just make this image in mind.
O is the orthocenter. Then angle BOC plus its opposite angle at the vertex. So angle BOC plus its opposite angle is angle A. Its sum will be 180 degree.
Very important point, angle BOC plus angle A is equal to 180 degree. Okay, what is angle BOC? angle made by orthocenter with a two vertices right so obviously third vertex will be left so that angle of orthocenter with the third vertex that is the opposite vertex that sum is always 180 degree right very important point okay now i can make such three again i can make another vertex here right so i can again make like okay see this is a triangle so again the triangle abc and now let's say again O is the orthocenter.
Okay, so now let's say I'm joining CO and AO. Right, so again this is the orthocenter. It is making this angle AOC with the other two vertices.
Right, so again what is the property? This angle of orthocenter plus its opposite angle at the vertex. Okay, so again this sum is what 180 degree. Fine, again that angle AO.
C plus angle B that sum is 180 degree right that means these two angles are supplementary okay, so angle made by also Center with the two vertices and Its opposite angle at the third vertex right both these angles are what supplementary supplementary means what sum of angles is 180 degree Okay, so very important point right very much this point very important. I can prove it also right it's quite easy to prove also Okay, you want me to prove it? I'll prove it right. But see, in geometry, we cannot prove every point, right? Because there are so many theorems here.
So many theorems, we are using so many results, right? So, it's not possible to prove every point. Otherwise, we study geometry for only 100 hours, right? So, I'll discuss all the points, whatever point is relevant for examinations, right? If you want any proof of it, you can just simply google it or search in NCR, Debook, right?
So, kindly, and I will... personally tell you to avoid the proofs, right? Don't always like, okay, this is a formula.
What is the proof, right? No, CAT and Z are aptitude examinations, right? They have to use that formula. That is like one concept, right?
This concept combined with other two, three concepts will form a single question. So, CAT is all about application of the concepts, right? So, especially in geometry.
So, here we are not needed to prove it, right? But okay. It's a small proof and it will help you also right so wherever it's required I'll obviously prove that results also right okay let me prove it We'll quickly make a triangle AB Quickly make a triangle AB let me draw all the altitudes now B is the altitude here C is the altitude here and A is the altitude here. Right, A is the altitude here. Fine.
Now see, this is the also center. Make it O. Okay. See. Now, look at triangle BC.
Look at triangle BC. Now I can write that, let this angle be theta. I am writing this angle as theta. I am marking this angle as phi.
Okay. This angle is theta and this angle is phi. Angle EB is theta.
Angle FCB is phi. Okay. Now, what is angle, look at triangle BC. In triangle BC, I can write like, Okay.
This theta is equal to, theta is equal to 90 minus angle C. Fine. I can write.
Because this is 90. This is, this whole is angle C. So obviously, this is the theta is the third angle of the triangle. For sum to be 180, if this angle is 90, this is theta. Oh sorry, this is angle C.
So what is theta? 90 minus angle C. Similarly, now look at triangle FCB.
Look at triangle FCB. In triangle FCB, again, sum of three angles. This angle is 90 degree.
Fine. This is phi and this is whole angle B. Whole angle B, right? So angle B plus angle phi plus 90 has to be?
180 degree because in triangle sum of three angles is 180 degree okay so here i can again write what is phi is equal to phi is equal to 90 minus angle b it is 90 minus angle b right now now look at triangle boc look at triangle boc i can write in triangle boc theta plus phi is equal to 180 minus angle boc i can write like this okay i can write like this that theta plus phi is equal to 180 minus angle Boc. Okay. Why?
Because, again, sum is 180 degree. So, theta plus phi plus angle B OC, sum has to be 180, right? Now, from these two equations, you can add and replace the value here, right? We have the value theta plus phi.
So, you can add these two equations. So, if you add these two equations, what is the result? Result will be theta plus phi will be equal to 180 minus angle B minus angle C. If I add these two equations, fine.
We'll put just theta plus phi here. So, by replacing again theta plus phi here, this equation looks like now, theta plus phi is 180 minus angle B minus angle C, right? Theta plus phi, I'm replacing with this, is equal to 180 minus angle BOC, okay? Is equal to 180 minus angle BOC.
Now, 180, 180 gets cancelled. That means angle BOC is equal to angle B plus angle C. And I can write angle BOC is equal to 180 minus angle A. Right, because in this bigger triangle AB, angle A plus angle B plus angle C will be equal to 180 degree. So, what is the value of angle B plus C?
So, B plus C will be what? 180 minus angle A. 180 minus angle A. It proves that angle BOC is equal to 180 minus angle A. Therefore, I can write therefore, I can write angle BOC plus angle A is equal to what?
180 degree, right? So, there is a proof for it, right? So, always remember any any and or So, the orthocenter angle plus its opposite vertex angle, sum is always 180. Right, we will do questions, right? So, we will get clarity there also. So, remember this.
Fine. Move to third geometric center. Okay.
Now, third geometric center is basically in-center. Third geometric center is what? In-center. Now, what is in-center?
In-center is basically intersection point of all the angle bisectors right so in center is basically intersection point of all the angle bisectors now what are bisectors bisector means cutting into equal halves right so obviously what are angle bisectors cut cutting a angle into two equal half right so let's draw that let me draw a triangle a b c So, at angle AB in this, let's say, now let's say A is the angle bisector. That means this angle and this angle is same. Now again, I can write B is the angle bisector of angle B. That means this angle and this angle is same.
And again, let's say C is the angle bisector of angle C. So, that means this angle and this angle are same, right? C is the angle bisector. Okay, now.
this again by construction for sure they'll meet at a common point okay and this common point is basically called the in center it is called the in center okay this common point is called the in center where they will intersect so it is in center okay now what is this uh property of incidental discuss right again there are few important things uh this relates to incident okay So, what is the in-center property here? So, in-center property here is C. Again, I can write like angle BIC is equal to 90 plus half of angle A.
Right? Similarly, angle AIB is equal to 90 plus half of angle C. and angle AIC is equal to 90 plus half of angle B, right?
There are three very important points, very important points, right? Again, I'll prove it, right? Because it's a small proof also and that concept we'll use further.
Okay, so I'll prove it. So, angle, first we can remember the results. So, it's easy to remember also what is angle BIC.
So, in angle BIC, this is angle BIC, right? It is 90 plus half of the opposite angle at the vertex, right? So angle B is the opposite angle is what?
Opposite angle is angle A. So angle B is equal to 90 plus half of angle A. Again what is angle AIC?
In angle AIC again 90 plus half of the opposite angle at the vertex. So 90 plus half of angle B. Fine. Now again let's prove it quickly.
You can prove it. See. Look at triangle BIC.
I can write that. In triangle BIC. In triangle BIC.
I can write. Now angle. Now since this is the angle bisector.
So again this angle is C by 2. This angle is C by 2. This angle is C by 2. This is C by 2 similarly. This is B by 2. This is b by 2, right? So, now you can see that in what is angle BIC?
Again, in triangle BIC, sum of these three angles will be equal to 180 degree, okay? So, that is angle BIC is equal to 180 minus b by 2 plus angle b by 2 plus angle c by 2. I can write this simply because sum has to be 180 degree, right? Let this be equation 1. Now, replace the value of angle B by 2 plus angle C by 2 by bigger triangle, right?
So, in bigger triangle AB, I can write. So, in bigger triangle AB, angle A plus angle B plus angle C is equal to 180 degree. Since I need to get the value of B by 2 and C by 2 because I am writing in terms of BIC only. Okay.
So, I have to replace this with. So, divide both sides by 2. So, angle A by 2 plus angle B by 2. plus angle c by 2 is equal to 90 degree angle c by 2 is equal to 90 degree okay 90 degree dividing both side by 2 so what is angle what is angle b by 2 plus angle c by 2 is 90 minus a by 2. you can replace here right it replaces here you simply put 180 minus 90 minus a by 2 is nothing but 90 plus angle a by 2 angle A by 2, right? So, you can see angle BIC is equal to 90 plus angle A by 2. This is the first result. Angle 90 plus angle A by 2. That is half of angle A, right? So, you can prove this.
It's a good result. And this is the property of in-center, right? Now, one more very important property of in-center lies here, right?
Let me rub it off. That from in-center, we can form an in-circle, right? And next point will be circumcenter.
So from circumcenter, we can form a circumcircle. Okay. So very important point, this one.
From in-center, we can form an in-circle, right? So let's again draw a triangle AB. So how to draw that in-circle C?
In triangle AB, let I be the in-center. Okay. So if I is the in-center, so keeping I as the center and keeping I as the center and taking one perpendicular on any side and keeping here the compass and keeping here the pencil compass here at eye and pencil here i can draw a circle like this which will touch the circle at exactly three points at exactly three points right this not perfect but also okay you understand it circle is touching ac that means ac is a tangent right This circle is called what?
This circle is called in circle. This circle is called in circle and its radius small r is called the in radius. Throughout geometry, we will use small r for in radius and capital R for circum radius. Circum radius we are going to discuss in next slide. Okay, so in circle and what is small r?
It is in radius. Okay, small r is what? It is in radius.
This is in circle, right? So, what is in circle? It will touch this circle at exactly three points from inside, right? Exactly three points.
This is the in circle. And its radius is called in radius, right? So, why did I make perpendicular here?
Why have I made perpendicular only here, right? Radius because B is the tangent to the circle, okay? Because B is the tangent to the circle and it is a property of circle that radius and tangent of circle always makes an angle of 90 degree. Okay, that means if this is the circle, okay, and this is the tangent, sorry, it's not a tangent, it was not touching it, right, so, and this is the tangent. So, touching the circle at just one point.
Okay, so radius and tangent makes an angle of what? 90 degree. Okay. So, again, it's a very important point, 90 degree.
So, radius tangent, 90 degree, right? That's why B is a tangent here. and this is the inner radius so this is the angle 90 degrees angle 90 degree okay so again a very important point with in center okay now next circumcenter next circumcenter fourth and last geometric center is circumcenter now again the root word right in center in in center it contains in in means what inside the circle is inside In circumcenter, it contains a root word circum. What is circum? Circum basically means outside.
Circum means outside, right? That means a circle will be formed outside the triangle, right? Okay, so what is circumcenter?
Circumcenter is nothing but intersection point of all the perpendicular side bisectors. all the perpendicular side bisectors, right? So, that basically means that if I draw a triangle here, triangle AB triangle AB and if this is a circumcenter, let this circumcenter here is represented by, again let's say O O is circumcenter here So, how to get a circumcenter?
Circumcenter will get by this This will be a perpendicular bisector of B. Right. Let's say this is the perpendicular bisector of B.
Okay. This line is making 90 degree as well as bisecting B. Okay.
Making 90 degree as well as bisecting B. Fine. Now, again, there will be a perpendicular bisector of AC as well. There will be a perpendicular bisector of AC as well.
Right. So again this line is making 90 degree with AC as well as bisecting in two equal halves. Then there is a perpendicular bisector of AB as well. So again this line will be like bisecting this AB and making 90. These two parts are equal right.
Diagram is a bit rough right so don't go by diagram. Okay this is 90 degree sorry this is circumspector. Intersection point of all the three. perpendicular side bisector perpendicular side bisector right this circumcenter what is the property property is that if this is circumcenter then keeping this point compass here and taking any one vertex as the radius this is the capital r right this capital r so i can i should write here that this is point is the circumcenter okay and capital R is the circum radius. Circum radius.
Okay. So, because circum center is, I should write here that circum center is, is equidistant, is equidistant from all three vertices. Right.
So, circum center is equidistant from all three vertices. Similarly, I should say in in center. In previous slides, I said in center, right? So, in center is equidistant from all the three sides. Should write here also.
I can write in center is equidistant from all three sides. Okay, why? Because if you draw this circle here, This is the in center, this is the in circle, right?
This is center, this perpendicular is also the radius, this perpendicular is also the radius and this perpendicular is also the radius, right? So all these three are equal, okay? That's why it is equidistant to all the three sides, fine? So keeping the compass at this point at O and taking any one vertex as radius, Since all the three vertices are at equal distance from the circumcenter, I have written here in the first point, I can draw a circle which will touch the circle triangle from exactly outside and at exactly three points. Right.
Exactly three points. Let me draw another one. Okay.
So exactly three points. Like this. Okay. This is what?
This is the circumcent, circle. Circumcircle means circum is outside. So, outside the circle. Outside the circle. right now one important property of circumcenter circum center is circum center if this o is a center here right if o is a center and if i join this c o join this c all right this is also circum radius and if i join this a o this is also circum radius right so if this angle a is theta understand right if this angle a is theta If this angle A is theta, then this angle BOC will be equal to 2 theta.
Right. If this angle A is theta, so this angle BOC is equal to 2 theta. Why?
Why? You can remember this property of circle. Right.
Let me remind you. The property of circle was. The property of circle was a chord. A chord which forms a certain angle at the center.
is double the angles formed by that chord at the circumference of the circle. Right? That means, let me write it into a diagram. So, in diagram I can make like, see.
In any circle, this is a circle here. Right? This is a chord.
Any line inside a circle is called a chord. Right? Any line inside the circle is called a chord. This is the center.
So, the first property is that if the angle formed at the vertex is theta. Okay. So, angle formed by the same chord at the center will be 2 theta.
Double of it. Right. This is what is depicted here. Right. Because this is the center of the circle.
This B is the chord here. B is the chord here. Okay. So, B is forming 2 theta at the center.
So, it will form theta at the circumference. Right. So, these two angles are just double of it. Right. Just double of it.
Okay. So, one more property again, circle again, just connect with right now is if AB is the chord, so AB in the same segment, right? This is one upper segment, this is one lower segment, right? So, in the same segment, all the angles formed by chord AB will be same. That means this angle is theta, this angle is also theta, okay?
This angle is also theta, right? So, all the angles are same, right? And at center.
it will form at center it will form 2 theta 2 theta right so very important points uh related to geometric center right these are the four genetic centers now we'll discuss its application some theorems in the next video okay thank you for watching geometry 2