1. Triangles: Points of Concurrency
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2. Investigate Points of Concurrency

3. Circumcenter

4. Perpendicular Bisectors and Circumcenters Examples

5. A perpendicular bisector of a triangle is a line or line segment that forms a right angle with one side of the triangle at the midpoint of that side. In other words, the line or line segment will be both perpendicular to a side as well as a bisector of the side. A B C D

6. Example 1: A F E C B D

7. Example 2: M Q N P

8. Since a triangle has three sides, it will have three perpendicular bisectors. These perpendicular bisectors will meet at a common point – the circumcenter. D
G is the circumcenter of ∆DEF.
Notice that the vertices of the triangle (D, E, and F) are also points on the circle. The circumcenter, G, is equidistant to the vertices. G E F

9. The circumcenter will be located inside an acute triangle (fig
The circumcenter will be located inside an acute triangle (fig.1), outside an obtuse triangle (fig. 2), and on a right triangle (fig. 3). In the triangles below, all lines are perpendicular bisectors. The red dots indicate the circumcenters.
fig. 1
fig. 3
fig. 2

10. Example 3: A company plans to build a new distribution center that is convenient to three of its major clients, as shown below. Why would placing this distribution center at the circumcenter be a good idea?

11. The circumcenter is equidistant to all three vertices of a triangle
The circumcenter is equidistant to all three vertices of a triangle. If the distribution center is built at the circumcenter, C, the time spent delivering goods to the three major clients would be the same. C

12. In Summary
The circumcenter is the point where the three perpendicular bisectors of a triangle intersect.
The circumcenter can be inside, outside, or on the triangle.
The circumcenter is equidistant from the vertices of the triangle

13. Circumcenter
Exploration
Construction

14. Try These: C

15. Try These: D

16. Try These: A

17. Try These B

19. Medians and Centroids Examples

20. A median of a triangle is a line segment that contains the vertex of the triangle and the midpoint of the opposite side. Therefore, the median bisects the side.

21. Since a triangle has three sides, it will have three medians
Since a triangle has three sides, it will have three medians. These medians will meet at a common point – the centroid.

22. The centroid is always located inside the triangle.
Acute triangle

23. The distance from any vertex to the centroid is 2/3 the length of the median.Q E F G S R D

24. Example 1: G is the centroid of triangle QRS. QG = 10 GF = 3
Example 1: G is the centroid of triangle QRS. QG = 10 GF = 3. Find QD and SF. Q E F G S R D

25. Example 3: G is the centroid of triangle DEF. FG = 15, ES = 21, QG = 5
Example 3: G is the centroid of triangle DEF FG = 15, ES = 21, QG = Determine FR, EG and GD E Q 5 15 G F R 21 S D

26. Notice that the distance from any vertex to the centroid is 2/3 the length of the median. That means that the distance from the centroid to the midpoint of the opposite side is 1/3 the length of the median. So, in triangle MNP, MQ=2(QT) and QT=(1/2)MQ M V N Q U T P

27. The centroid is also known as the balancing point (center of gravity) of a triangle.

28. In Summary
A median is a line segment from the a vertex of a triangle to the midpoint of the opposite side.
The distance from the vertex to the centroid is 2/3 the length of the median.
The distance from the centroid to the midpoint is 1/3 the length of the median, or half the distance from the vertex to the centroid.
Since the centroid is the balancing point of the triangle, any triangular item that is hung by its centroid will balance.

29. Centroids
Investigate
Construction

30. Try These: D

31. Try These: C

32. Try These: A

33. Try These: C

35. Angle Bisectors and Incenters Examples

36. An angle bisector of a triangle is a segment that shares a common endpoint with an angle and divides the angle into two equal parts.

37. Example 1: Determine any angle bisectors of triangle ABC.

38. Since a triangle has three angles, it will have three angle bisectors
Since a triangle has three angles, it will have three angle bisectors. These angle bisectors will meet at a common point – the incenter. X M Z Y

39. The incenter is always located inside the triangle.
Acute triangle
Obtuse triangle
Right triangle

40. The incenter is equidistant to the sides of the triangle.x x

41. Example 2: L is the incenter of triangle ABC
Example 2: L is the incenter of triangle ABC. Which segments are congruent? A D B L E F C

42. Example 3: Given P is the incenter of triangle RST
Example 3: Given P is the incenter of triangle RST. PN = 10 and MT = 12, find PM and PT. 12 10
Not drawn to scale

43. In Summary
The incenter is the point of intersection of the three angle bisectors of a triangle.
The incenter is equidistant to all three sides of the triangle.

44. Incenter
Investigate
Construction

45. Try These: B

46. Try These: C

47. Try These: C

48. Try These: K C

50. Altitudes and Orthocenters Examples

51. An altitude of a triangle is the perpendicular distance between a vertex and the opposite side. This distance is also known as the height of the triangle. A B D C

52. Example 1: Determine any altitudes of triangle ABC.

53. Since a triangle has three sides, it will have three altitudes
Since a triangle has three sides, it will have three altitudes. These altitudes will meet at a common point – the orthocenter. X O Y Z

54. The orthocenter may be located inside an acute triangle (fig
The orthocenter may be located inside an acute triangle (fig. 1), outside an obtuse triangle (fig. 2), or on a right triangle (fig. 3). In the triangles below, the red lines represent altitudes. The red points indicate the orthocenters.
Obtuse Triangle
Acute Triangle
Right Triangle
Fig. 3
Fig. 2
Fig. 1

55. Summary
An altitude is a line segment containing a vertex of a triangle and is perpendicular to the opposite side.
The orthocenter is the intersection point of the three altitudes of a triangle.
Orthocenters can be inside, outside, or on the triangle depending on the type of triangle.

56. Orthocenter
Investigate
Construction

57. Try These: B

58. Try These: B

59. Try These: A

60. Points of Concurrency
Investigate

61. Points of Concurrency
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Review

62. Try These: A

63. Try These: B

64. Try These: D

65. Try These: B

66. Try These: A

67. Try These: A

68. Try These: B

69. Try These: D

70. Try These: A

71. Try These: C