Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles

1
Chapter 2 Circumcenter, Orthocenter,
incenter, and centroid of triangles

• Outline
• Perpendicular bisector ,
• circumcentre and orthocenter
• Bisectors of angles and the incentre
• Medians and centroid

2
2.1 Perpendicular bisector, Circumcenter and
orthocenter of a triangle

• Definition 1 The perpendicular bisector
• of a line segment is a line perpendicular
• to the line segment at its midpoint.

CD is a perpendicular bisector of AB if (i)
ACBC (ii) DCA DCB
3
In-Class-Activity 1

• (1) If P is a point on the perpendicular
  bisector of AB, what is the relationship
  between PA and PB?
• (2) Make a conjecture from the observation in
  (1). Prove the conjecture.
• (3) What is the converse of the conjecture in
  (2).
• Can you prove it?

4

• Theorem 1 The perpendicular bisectors of the
  three sides of a triangle meet at a point
• which is equally distant from the vertices
  of the triangle.

The point of intersection of the three
perpendicular bisectors of a triangle is called
the circumcenter of the triangle.
5

• DG, MH and EF are the perpendicular
  bisectors of the sides AB,AC and BC
  respectively
• DG, MH and EF meet at a point O
• OAOBOC
• O is the circumcenter of triangle ABC.

6
Proof of Theorem 1

• Given in ABC that DG, EF and MH are the
  perpendicular bisectors of sides AB, BC and CA
  respectively.
• To prove that
• DG,EF and MH meet at a point O,
• and AOBOCO.
• Plan Let DG and EF meet at a point O. Then show
  that OM is perpendicular to AC.

7
Proof

• 1.Let DG and EF meet at O
• 2. Connect M and O.
• We show MO is
• perpendicular to side AC
• 3. Connect AO, BO and CO.

• (If they dont meet, then DG//EF, so AB//BC,
  impossible)

8

• 4. AOBO, BOCO
• 5. AOCO
• 6. MOMO
• 7. AMCM
• 8.
• 9.
• 10

• (O is on the perpendicular bisects of AB and BC)
• ( By 4 )
• (Same segment )
• ( M is the midpoint )
• (S.S.S)
• (Corresponding angles
• (By 9 and )

9

• 11.OM is the perpendicular (Two conditions
  satisfied)
• bisector of side AC.
• 12. The three perpendicular
• bisector meet at point O.
• 13.O is equally distant from ( by 4)
• vertices A,B and C.

10

• Remark 1 ( A method of proving that three lines
  meet at a point )
• In order to prove three lines meet at one point,
  we can
• first name the meet point of two of the lines
• then construct a line through the meet point
• (iii) last prove the constructed line coincides
  with the third line.

11

• In-Class-Exercise 1
• Prove Theorem 1 for obtuse triangles.
• Draw the figure and give the outline of the
• proof

12

• Remark 2 The circumcenter of a triangle is
  equally distant from the three vertices.
• The circle whose center is the
  circumcenter of a triangle and whose radius is
  the distance from the circumcenter to a vertex
  is called the
• circumscribed circle
• of the triangle.

13

• In-Class-Activity
• Give the definition of parallelograms
• (2) List as many as possible conditions for a
  quadrilateral to be a parallelogram.
• (3) List any other properties of parallelogram
  which are not listed in (2).

14
(1) Definition A parallelogram is a quadrilateral
with its opposite sides parallel
ABCD

• (2) Conditions
• The opposite sides equal
• Opposite angles equal
• The diagonals bisect each other
• Two opposite side parallel and equal
• (3)

15

• Theorem 2
• The three altitudes of a triangle meet at a
  point.

16

• Given triangle ABC with altitudes AD, BE
  and CF.
• To prove that AD, BE and CF meet at a point.
• Plan is to construct another larger triangle
  ABC
• such that AD, BE and CF are the perpendicular
  bisectors
• of the sides of ABC. Then apply
  Theorem 1.

17

• Proof (Brief)
• Construct triangle ABC such that
• AB//AB, AC//AC, BC//BC
• 1. ABCB is a parallelogram.
• 2. BCAB.
• 3. Similarly CAAB.
• 4. CE is the perpendicular bisector of ABC
  of side BA.
• 5. Similarly BF and AD are perpendicular
  bisectors of sides of ABC.
• 6. So AD, BF and CE meet at a point (by
  Theorem 1)

18

• The point of intersection of the three
  altitudes of a triangle is called the
• orthocenter
• of the triangle.

19
2.2 Angle bisectors , the incenter of a
triangle

• Angle bisector
• ABD DBC
• In-Class-Exercise 2
• (1) Show that if P is a point on the bisector
  of then the distance from P to AB
  equals the distance
• from P to CB.
• (2) Is the converse of the statement in (1)
  also true?

20

• Lemma 1 If AD and BE are the bisectors of the
  angles
• A and B of ABC, then AD and BE
  intersect at a point.

Proof Suppose they do not meet. 1. A
B C180 ( Property of triangles) 2.
Then AD// BE. ( Definition of parallel
lines) 3. DAB EBA180 ( interior
angles on same side ) 4.
( AD and BE are bisectors
)

21

• 5.This contradicts that
• The contradiction shows that the two angle
  bisectors must meet at a point.

Proof by contradiction ( Indirect proof) To
prove a statement by contradiction, we first
assume the statement is false, then deduce
two statements contradicting to each other.
Thus the original statement must be true.
22

• Theorem 3 The bisectors of the three angles of
  a triangle
• meet at a point that is equally distant from the
  three side
• of the triangle.

The point of intersection of angle bisectors of
a triangle is called the
incenter of the triangle
Read and complete the proof
23

• Remark Suppose r is the distance from the
  incenter to a side of a triangle. Then there
  is a circle whose center is the incenter and
  whose radius is r.
• This circle tangents to the three sides
• and is called the
• inscribed circle ( or incircle) of the
  triangle.

24
Example 1 The sum of the distance from any
interior point of an equilateral triangle to the
sides of the triangle is constant.
25

• Proof
  
• 1.
• 2.
• 3. ABACBC (ABC is equilateral )
• 4.
• 5.
  ( by 1 and 4)
• 6.
• is a constant.

26

• In-Class-Activity
• (1) State the converse of the conclusion proved
  in Example 1.
• Is the converse also true?
• Is the conclusion of Example 1 true for points
  outside the triangle?

27
2.3 Medians and centroid of a triangle

• A median of a triangle is a line drawn from any
  vertex to the mid-point of the opposite side.
• Lemma 2 Any two medians of a triangle meet at a
  point.

28

• Theorem 3 The three medians of a triangle meet
  at a point which is two third of the distance
  from each vertex to the mid-point of the opposite
  side.

The point of intersection of the three medians of
a triangle is called the
centroid of the triangle
29

• Proof (Outline)
• Let two median AD and BE meet at O.
• Show
• If CE and AE meet at O, then
• So O is the same as O
• All medians pass through O.
• Read the proof

30

• Example 2 Let line XYZ be parallel to side
  BC and pass
• through the centroid O of .
• BX, AY and CZ are perpendicular to XYZ.
• Prove AYBXCZ.

31

• .

32
Question

• Is the converse of the conclusion in
• Example 2 also true?
• How to prove it?

33

• Summary
• The perpendicular bisectors of a triangle meet at
  a point---circumcenter, which is equally distant
  from the three vertices and is the center of the
  circle outscribing the
• triangle.
• The three altitudes of a triangle meet at a
  point--- orthocenter .
• The angle bisectors of a triangle meet at a
  point---incenter, which is equally distant from
  the three sides and is the center of the circle
  inscribed the triangle.
• The three medians of a triangle meet at a point
  ---centroid. Physically, centroid is the center
  of mass of the triangle with uniform density.

34
Key terms

• Perpendicular bisector
• Angle bisector
• 
  Altitude
• 
  Median
• Circumcenter
• 
  Orthocenter
• Incenter
• 
  Centroid
• Circumscribed circle
• Incircle

35
Please submit the solutions of 4
problems in Tutorial 2
next time. THANK YOU
Zhao Dongsheng MME/NIE Tel [phone] E-mail
dszhao_at_nie.edu.sg