Bisectors of Triangles // GEOMETRY

Summary

In this educational video, Gary LS from Gary Green explains the concept of 'concurrency' in triangles, focusing on the points where lines intersect. He introduces the 'circumcenter' as the point where the perpendicular bisectors of a triangle meet, equidistant from all vertices. The video also covers the 'incenter,' the point of concurrency for angle bisectors, which is always inside the triangle and equidistant from all sides. The content is presented in an engaging manner, with a touch of humor, to help viewers grasp these geometric concepts.

Takeaways

• 📌 The term 'concurrent' refers to three lines intersecting at a single point, known as the 'point of concurrency'.
• 📐 When drawing perpendicular bisectors in a triangle, they always intersect at a single point called the 'circumcenter'.
• 🔄 The circumcenter is equidistant from all the vertices of the triangle, regardless of the triangle's type.
• 🏞️ In an acute triangle, the circumcenter is located inside the triangle, while in an obtuse triangle, it is outside.
• 📍 For a right triangle, the circumcenter lies on the hypotenuse.
• 🌐 The circumcenter can be used to circumscribe a circle around the triangle, touching all vertices at equal distances.
• 🔶 The 'incenter' is the point of concurrency for the angle bisectors of a triangle and is always located inside the triangle.
• 📏 The incenter is equidistant from all the sides of the triangle, not the vertices, and this distance is measured perpendicularly to the sides.
• 🌀 A circle can be inscribed within the triangle that touches each side at the point equidistant from the incenter.
• 📝 The concept of 'concurrency' is central to understanding the properties of the circumcenter and incenter in the context of triangles.
• 🎨 The video script emphasizes the importance of visual representation and understanding geometric concepts through drawing and visualization.

Q & A

What does the term 'concurrent' refer to in the context of lines?

In the context of lines, 'concurrent' refers to three or more lines that intersect at a single point, which is known as the point of concurrency.

What is the special name given to the point where the perpendicular bisectors of a triangle meet?

The point where the perpendicular bisectors of a triangle meet is called the circumcenter.

What is the circumcenter's relationship to the vertices of a triangle?

The circumcenter is equidistant from all the vertices of the triangle.

Where is the circumcenter located in relation to the triangle for different types of triangles?

For an acute triangle, the circumcenter is inside the triangle. For an obtuse triangle, it is outside the triangle. For a right triangle, it lies on the hypotenuse.

What is meant by a circle being circumscribed around a triangle?

A circle is said to be circumscribed around a triangle if it touches all the vertices of the triangle, with the circumcenter being the center of this circle.

What is the point of concurrency called when the angle bisectors of a triangle meet?

When the angle bisectors of a triangle meet, the point of concurrency is called the incenter.

Is the incenter always inside the triangle?

Yes, the incenter is always located inside the triangle.

What is unique about the incenter's distance to the sides of the triangle?

The incenter is equidistant from all the sides of the triangle, meaning it is the same distance from each side when measured perpendicularly.

How does the incenter relate to the perpendicular bisectors of the triangle?

The incenter is the point where the angle bisectors of the triangle meet, and it is equidistant from the perpendicular bisectors, not the sides.

What is the significance of the incenter in terms of balancing the triangle?

The incenter is significant for balancing the triangle because it is the point where the triangle can be balanced on a pin, as it is the center of the inscribed circle that touches all three sides.

Outlines

00:00

📐 Understanding Triangle Bisectors and Points of Concurrency

The script introduces the concept of concurrent lines in triangles, where three lines intersect at a single point called the point of concurrency. It explains the perpendicular bisectors of a triangle, which intersect at the circumcenter, a point equidistant from all vertices. The video distinguishes the circumcenter's position based on the type of triangle: inside for acute, outside for obtuse, and on the hypotenuse for right triangles. It also emphasizes the circumcenter's role in circumscribing a circle around the triangle, touching all vertices at equal distances.

05:00

🔄 Exploring the Incenter and its Properties

This paragraph delves into the angle bisectors of a triangle, which intersect at the incenter, a point always located inside the triangle. The incenter is unique in that it is equidistant from all sides of the triangle, not the vertices. The script illustrates this by showing that perpendicular lines drawn from the incenter to each side are of equal length. The incenter is also the center of a circle inscribed within the triangle, touching each side at its midpoint. The summary highlights the importance of understanding the incenter's role in balancing and symmetry within the triangle.

Keywords

💡Concurrent

Concurrent refers to the condition where three or more lines intersect at a single point. In the context of the video, it is essential for understanding the concept of points of concurrency within triangles. The script uses the term to describe the meeting point of the perpendicular bisectors of a triangle's sides, which is a central theme of the video.

💡Point of Concurrency

The point of concurrency is the specific location where concurrent lines intersect. In the video, this term is used to identify the point where the perpendicular bisectors of a triangle's sides meet, which is a key concept in discussing the circumcenter and incenter of a triangle.

💡Perpendicular Bisectors

Perpendicular bisectors are lines that are both perpendicular to a segment and bisect it into two equal parts. The video explains that the perpendicular bisectors of a triangle's sides always meet at a single point, known as the circumcenter, which is central to the discussion of the triangle's geometry.

💡Circumcenter

The circumcenter is the point where the perpendicular bisectors of a triangle's sides intersect. It is equidistant from all the triangle's vertices. The video emphasizes the circumcenter's significance by explaining its location relative to the triangle's sides and vertices, and how it varies with different types of triangles (acute, obtuse, right).

💡Equidistant

Equidistant describes a point that is at the same distance from multiple other points. In the video, the term is used to describe the circumcenter's property of being the same distance from all the triangle's vertices, as well as the incenter's property of being the same distance from all the triangle's sides.

💡Acute Triangle

An acute triangle is a triangle where all angles are less than 90 degrees. The script mentions that the circumcenter of an acute triangle is located inside the triangle, which is an important characteristic when discussing the properties of the circumcenter.

💡Obtuse Triangle

An obtuse triangle has one angle greater than 90 degrees. The video points out that the circumcenter of an obtuse triangle is located outside the triangle, which contrasts with the location of the circumcenter in acute triangles.

💡Right Triangle

A right triangle has one angle exactly equal to 90 degrees. The script specifies that the circumcenter of a right triangle lies on the hypotenuse, which is a unique property among the types of triangles discussed.

💡Incenter

The incenter is the point of concurrency for the angle bisectors of a triangle. It is always located inside the triangle and is equidistant from all the triangle's sides. The video explains the incenter's importance by describing its geometric properties and its role in balancing the triangle.

💡Angle Bisectors

Angle bisectors are lines that divide an angle into two equal parts. The video uses the concept of angle bisectors to introduce the incenter, explaining that the incenter is where all the angle bisectors of a triangle intersect, which is a fundamental aspect of the triangle's geometry.

Highlights

Gary LS introduces the concept of concurrent lines and the point of concurrency in a triangle.

Explanation of the term 'concurrent' and its significance in geometry.

Introduction to the concept of perpendicular bisectors and their role in triangle geometry.

The point where perpendicular bisectors meet is called the circumcenter of a triangle.

The circumcenter is equidistant from all vertices of the triangle.

Location of the circumcenter varies depending on the type of triangle: inside for acute, outside for obtuse, and on the hypotenuse for right triangles.

Circumcenter can be used to circumscribe a circle around a triangle, touching all vertices.

Transition to discussing angle bisectors and their significance in triangle geometry.

The point where angle bisectors meet is called the incenter of a triangle.

The incenter is always located inside the triangle and is equidistant from all sides.

The incenter is used to inscribe a circle within a triangle, touching all sides.

Difference between the circumcenter and incenter in terms of their distances from vertices and sides.

The incenter's unique property of being equidistant to all sides, requiring perpendicular lines to the sides.

Gary LS emphasizes the importance of understanding the concepts of circumcenter and incenter for geometric problem-solving.

Summary of the key differences between the circumcenter and incenter in a triangle.

Final review of the concepts discussed, reinforcing the understanding of triangle geometry related to concurrency.

Transcripts

[Music]

all right ladies and gentlemen today's

message is brought to you by Gary LS of

Gary green for all your suing needs all

right we're doing 5-2 bis sectors of

triangles all right bis sectors of

triangles this one's a doozy so take

some notes and Rewind it if you need to

stop it if you have to and just get

another teacher if I'm a Crum one all

right first word need to know is

concurrent so concurrent

that means that three

lines intersect that makes them

concurrent now that point right there

that little spot where they all hit is

called the point of

concurrency I'm be using that term quite

a bit so be good to pay attention to

that so we got concurrent and then point

of

concurrency all right now when we

talking about stuff with triangles a lot

but they all have points of concurrency

but but they all have different names

for that okay

on all right here we

go

R gosh that's an awful I've never do

that again all right let's try this

triangle right here say I want to draw

all three perpendicular bis sectors okay

all the perpendicular bis sectors

perpendicular means they're going to be

in the middle and it's going to be per

that's bisect and then that's

perpendicular so it cuts that in half

and that's perpendicular let's see

Cuts this in half right there I'd

say

perpendicular and then this one would be

right

here all right guess what that's

perpendicular as well they're always

going to hit when you do the

perpendicular bis

sectors for any triangle they're always

going to hit at the same spot that's

called their point of concurrency now

there's a special name for that okay

when the perpendicular bis sectors of a

triangle meet it is

called the circum Center if I'm saying

it right yep

circumcenter it's called the

circumcenter of the triangle now that's

only when the perpendicular bis sectors

all hit okay that's the point of

concurrency it's called the circumcenter

there's going to be a different name for

different stuff when we do other things

okay but that's what you need to know is

that's circumcenter

all right

now

um okay now this circum center right

here it is the exact same distance to

all the vertices like this

line this line and this line I didn't

draw that last one very

well those are all the exact same

distance

I know there's a lot of stuff going on

here but what you need to know is if you

have a triangle you do the perpendicular

bis sectors they all cross at one point

that's called the circum Center okay

that circumcenter is the same distance

to all the vertices to all the angles

okay goody goody gumdrops all right

let's move on to the next

one uh oh side note for circumcenter if

it's an acute triangle that point is

going to be inside the tri triangle if

it's an obtuse triangle it'll be outside

the triangle and if it's a right

triangle it's going to be on the

hypotenuse so some fun facts you can

take with you on your vacation okay

now now the circum Center we call it the

circum Center it is always circumscribed

inside the triangle circumscribed or uh

yeah circumscribes what it's called make

sure I don't say something wrong

everybody's going to make fun of me all

right what you do is that means that if

I made like a circle A Perfect Circle

that touched all these vertices which I

know I'm going to mess this

up all right let's pretend that that was

a perfect circle that circum Center it

would be in the exact middle of my

circle same distance to everything all

right I'm making it look more compli

because I thought it would look pretty

but it turns out it just looks like an

ugly triv Pursuit piece all right so

that's the circumcenter all right main

thing you need to know is the

circumcenter is where the perpendicular

bis sectors meet and it's equal distant

to all the

vertices goodness gracious I said that

like 19 times okay so we got perp

perpendicular bis sectors out of the way

now that was the three perpendicular bis

sectors next is the three angle bis

sectors okay

now I know you probably think that I

just edited that video because I went so

fast you probably couldn't even see me

it was like a blur but I didn't it was

on me took some n n classes in high

school don't worry about

it okay so the next one let's say we

have another

triangle okay that looks like a right

triangle but don't matter whatever I'll

do whatever I want all right first thing

we did was where the perpendicular bis

sectors meet now we're doing it where

the angle bis sectors meet

okay angle bis sectors cut this angle in

half looks something like that that one

looks something like that and this one

looks something like

this those are all angle bisector say I

cut that in half I cut that in half and

I cut that in

half okay right there where they all

meet their point of

concurrency is called the N

Center okay it's called the in center

and guess what it's in the center all

right like if you want to to balance

this full like like if I had a triangle

and I was trying to balance it on a pin

I would probably put it at the end

Center okay but that's just me I'm just

one man all right

now now uh in Center let me see what I'm

not telling you in Center is always

going to be inside the triangle always

you know circumcenter it was inside

outside or on the hypotenuse in Center

is always inside the triangle now also

you know how last time we put a triangle

uh we put a circle that went all the way

around it this time if we put a circle

inside of

it that touched every single one of

every single one of like the

perpendicular bis sectors it would make

it where that's the exact center it

doesn't look like it right there because

I'm a horrible artist I have no skills

of an artist and let's see there's one

more thing I need to tell you about in

center and I'll tell you right now all

right the in center of a tri triangle

you know how last time we did a

perpendicular bis sector thing and we

found the circumcenter and it was equal

distance equidistant to all the vertices

this one is equidistant to all the sides

if you go straight there make that

perpendicular make that

perpendicular make that perpendicular

that would be all the same length right

there those three right there okay and I

know it looks crappy cuz I have a lot of

stuff in here and I'm sorry please

forgive me

all

right but it's the same distance to all

those all right so perf I'm just going

keep reviewing and going back that way

you know first thing we did all the

perpendicular bis sectors their point of

concurrency is called the circum Center

okay it is equidistant to all the angles

the vertices this one angle bis sector

it uh is where you go from the angle B

sectors their point of concurrency is

always inside the triangle it is called

the in center and is equal distant to

all the sides but you have to go

straight to the sides remember what we

did last chapter talking about the

shortest distance to any side makes

perpendicular line yeah all right um I'm

out of breath so I think I'm just going

to end this one right here