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An example on Given O(0, 0), A(12, 0), B( The aim of this small article i co-ordinates of the five classi Δ OAB and other related poin We choose a right-angled trian simplicity. (1) Orthocenter: The three altitudes of a tri one point called the orthoc (Altitudes are perpendicul vertices to the opposite sid triangles.) If the triangle is obtuse, th the orthocenter is the verte Conclusion : Simple, the o (2) Circum-center: The three perpendicular b a triangle meet in one poin circumcenter. It is the cent the circle circumscribed ab If the triangle is obtuse, th outside the triangle. If it is the circumcenter is the mid hypotenuse. (By the theore semi-circle as in the diagra Conclusion : the circum n five classical centres of a right angled trian (0, 5) . is to find the ical centres of the nts of interest. angle for iangle meet in center. lar lines from des of the he orthocenter is outside the triangle. If it ex which is the right angle. orthocenter = O(0, 0) . bisectors of the sides of nt called the ter of the circumcircle, bout the triangle. hen the circumcenter is s a right triangle, then dpoint of the em of angle in am.) m-centre, E = , 6, 2.5 1 ngle Yue Kwok Choy t is a right triangle,

Transcript of five Classical Centres - Queen's College On The Web example on five classical... · co-ordinates of...

An example on five classical centres of a right angled triangle

Given O(0, 0), A(12, 0), B(0, 5) .

The aim of this small article is to find the

co-ordinates of the five classical centres

∆ OAB and other related points of interest.

We choose a right-angled triangle for

simplicity.

(1) Orthocenter:

The three altitudes of a triangle meet in

one point called the orthocenter.

(Altitudes are perpendicular lines from

vertices to the opposite sides of the

triangles.)

If the triangle is obtuse, the orthocenter is outside the triangle. If it is a right triangle,

the orthocenter is the vertex which is the right angle.

Conclusion : Simple, the orthocenter

(2) Circum-center:

The three perpendicular bisectors

a triangle meet in one point called the

circumcenter. It is the center of the circumcircle,

the circle circumscribed about the triangle.

If the triangle is obtuse, then the circumcenter is

outside the triangle. If it is a right triangle, then

the circumcenter is the midpoint of the

hypotenuse. (By the theorem of angle in

semi-circle as in the diagram.)

Conclusion : the circum

An example on five classical centres of a right angled triangle

O(0, 0), A(12, 0), B(0, 5) .

is to find the

classical centres of the

OAB and other related points of interest.

angled triangle for

of a triangle meet in

one point called the orthocenter.

(Altitudes are perpendicular lines from

vertices to the opposite sides of the

If the triangle is obtuse, the orthocenter is outside the triangle. If it is a right triangle,

the orthocenter is the vertex which is the right angle.

orthocenter = O(0, 0) .

perpendicular bisectors of the sides of

a triangle meet in one point called the

circumcenter. It is the center of the circumcircle,

the circle circumscribed about the triangle.

triangle is obtuse, then the circumcenter is

outside the triangle. If it is a right triangle, then

the circumcenter is the midpoint of the

(By the theorem of angle in

circle as in the diagram.)

the circum-centre, E = ������ , ���

� � 6,2.5�

1

An example on five classical centres of a right angled triangle

Yue Kwok Choy

If the triangle is obtuse, the orthocenter is outside the triangle. If it is a right triangle,

Exercise 1:

(a) Check that the circum

(b) Show that the area of the triangle

����� 2R� sinA sin

(3) Centroid:

The three medians (the lines drawn

from the vertices to the bisectors of

the opposite sides) meet in the

centroid or center of mass. The

centroid divides each median in a

ratio of 2 : 1.

Since OR : RA = 1 : 1, we have R =

Since BC : CR = 2 : 1, we

Conclusion : Centroid,

Exercise 2: Prove that i

coordinates of the centroid of

(4) In-center: The three angle

bisectors of a triangle meet in one

point called the in-center. It is the

center of the in-circle, the circle

inscribed in the triangle.

Let OA = a = 12,

OB = b = 5

AB = c = √12� � 5�

Semi-perimeter, s =

The radius of the incircle = r = EX = EY = EZ

Then Area of ∆OAB = Area of

��� ��

� � ��� � ��

∴ r ���! ��"�

�"��

that the circum-circle above is given by: x $ 6�� � he area of the triangle with sides a, b, c and angles A, B, C

sinB sin C , where R is the radius of the circum

(the lines drawn

from the vertices to the bisectors of

the opposite sides) meet in the

centroid or center of mass. The

centroid divides each median in a

Since OR : RA = 1 : 1, we have R = 6, 0�. Since BC : CR = 2 : 1, we therefore have

Centroid, C = ��"(��"���� , �"���"�.�

��� � �4, �*� Prove that if A x�, y��, B x�, y��, C x

coordinates of the centroid of ∆ABC is given by �,-�,.�,*

angle

of a triangle meet in one

center. It is the

circle, the circle

� 13

s = �����

� ������*� 15

The radius of the incircle = r = EX = EY = EZ

OAB = Area of ∆EOA + Area of ∆EOB+ Area of

��� ������

� � r sr 2

2

y $ 2.5�� 6.5�. with sides a, b, c and angles A, B, C is

where R is the radius of the circum-circle .

� . x*, y*� , then the

,/ , 0-�0.�0/* � .

EOB+ Area of ∆EAB

Conclusion : In-centre

In-circle :

Exercise 3:

Prove that the radius of a general

semi-perimeter s is given by :

(5) Ex-center: An ex-circle of the triangle is a

circle lying outside the triangle, tangent to

one of its sides and tangent to the

extensions of the other two.

Every triangle has three distinct ex

each tangent to one of the triangle's sides.

The center of an ex-circle is t

of the internal bisector of one angle and the

external bisectors of the other two.

As in the diagram,

Area of ∆OAB

= Area of ∆E1AB + ∆E1OB

ab2 cr�

2 � br�2 $

4������� �

∴ r� ���!5��= ��

���

Similarly, r� ���!5��=

Conclusion : Ex-centre

Exercise 4:

(a) Write down the equations of the ex

(b) Prove that for a general triangle (not just right) the radii of the ex

r� 6!!5��!5��!5� ,r

centre : E = (r, r) = (2, 2)

circle : x $ 2�� � y $ 2�� 2�

Prove that the radius of a general ∆ABC (not just right) with sides a, b, c and

perimeter s is given by : r 6!5��!5��!5��!

of the triangle is a

circle lying outside the triangle, tangent to

one of its sides and tangent to the

extensions of the other two.

Every triangle has three distinct ex-circles,

each tangent to one of the triangle's sides.

circle is the intersection

of one angle and the

of the other two.

OB - ∆E1OA

$ ar�2 7b � c $ a

2 8 r� 7a � b � c $2

� $ a9 r� s $ a�r� ��"���5��� 10 =

��"����5�� 3, r* ��

�!5��= ��"����5�*� 15

centres : E� r�, $r�� 10,$10� E� $r�, r�� $3,3� E* r*, r*� 15,15�

the equations of the ex-circles in the above.

Prove that for a general triangle (not just right) the radii of the ex

r� 6!!5��!5��!5� ,r* 6!!5��!5��

!5�

3

with sides a, b, c and

2a8 r�

15

Prove that for a general triangle (not just right) the radii of the ex-circles are