1. Introduction
Think about all the properties of triangles we have learned so far and all the constructions we are able to perform. What properties exist when the perpendicular bisectors of triangles are constructed? Is there anything special about where the angle bisectors of a triangle intersect? We know triangles have three altitudes, but can determining each one serve any other purpose? How can the midpoints of each side of a triangle help find the center of gravity of a triangle? Each of these questions will be answered as we explore the centers of triangles.
1.9.4: Proving Centers of Triangles

2. Key Concepts Every triangle has four centers.
Each center is determined by a different point of concurrency—the point at which three or more lines intersect.
These centers are the circumcenter, the incenter, the orthocenter, and the centroid.
1.9.4: Proving Centers of Triangles

3. Key Concepts, continued
Circumcenters
The perpendicular bisector is the line that is constructed through the midpoint of a segment. In the case of a triangle, the perpendicular bisectors are the midpoints of each of the sides.
The three perpendicular bisectors of a triangle are concurrent, or intersect at one point.
1.9.4: Proving Centers of Triangles

4. Key Concepts, continued
This point of concurrency is called the circumcenter of the triangle.
The circumcenter of a triangle is equidistant, or the same distance, from the vertices of the triangle. This is known as the Circumcenter Theorem.
1.9.4: Proving Centers of Triangles

5. Key Concepts, continued
Theorem
Circumcenter Theorem
The circumcenter of a triangle is equidistant from the vertices of a triangle.
The circumcenter of this triangle is at X.
1.9.4: Proving Centers of Triangles

6. Key Concepts, continued
The circumcenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle.
The circumcenter is inside acute triangles, outside obtuse triangles, and on the midpoint of the hypotenuse of right triangles.
1.9.4: Proving Centers of Triangles

7. Key Concepts, continued
Look at the placement of the circumcenter, point X, in the following examples.
Acute triangle
Obtuse triangle
Right triangle
X is inside the triangle.
X is outside the triangle.
X is on the midpoint of the hypotenuse.
1.9.4: Proving Centers of Triangles

8. Key Concepts, continued
The circumcenter of a triangle is also the center of the circle that connects each of the vertices of a triangle. This is known as the circle that circumscribes the triangle.
1.9.4: Proving Centers of Triangles

9. Key Concepts, continued
Incenters
The angle bisectors of a triangle are rays that cut the measure of each vertex in half.
The three angle bisectors of a triangle are also concurrent.
This point of concurrency is called the incenter of the triangle.
The incenter of a triangle is equidistant from the sides of the triangle. This is known as the Incenter Theorem.
1.9.4: Proving Centers of Triangles

10. Key Concepts, continued
Theorem
Incenter Theorem
The incenter of a triangle is equidistant from the sides of a triangle.
The incenter of this triangle is at X.
1.9.4: Proving Centers of Triangles

11. Key Concepts, continued The incenter is always inside the triangle.
Acute triangle
Obtuse triangle
Right triangle
1.9.4: Proving Centers of Triangles

12. Key Concepts, continued
The incenter of a triangle is the center of the circle that connects each of the sides of a triangle. This is known as the circle that inscribes the triangle.
1.9.4: Proving Centers of Triangles

13. Key Concepts, continued
Orthocenters
The altitudes of a triangle are the perpendicular lines from each vertex of the triangle to its opposite side, also called the height of the triangle.
The three altitudes of a triangle are also concurrent.
This point of concurrency is called the orthocenter of the triangle.
1.9.4: Proving Centers of Triangles

14. Key Concepts, continued
The orthocenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle.
The orthocenter is inside acute triangles, outside obtuse triangles, and at the vertex of the right angle of right triangles.
1.9.4: Proving Centers of Triangles

15. Key Concepts, continued
Look at the placement of the orthocenter, point X, in the following examples.
Acute triangle
Obtuse triangle
Right triangle
X is inside the triangle.
X is outside
the triangle.
X is at the vertex of the right angle.
1.9.4: Proving Centers of Triangles

16. Key Concepts, continued
Centroids
The medians of a triangle are segments that join the vertices of the triangle to the midpoint of the opposite sides.
Every triangle has three medians.
The three medians of a triangle are also concurrent.
1.9.4: Proving Centers of Triangles

17. Key Concepts, continued
This point of concurrency is called the centroid of the triangle.
The centroid is always located inside the triangle the distance from each vertex to the midpoint of the opposite side. This is known as the Centroid Theorem.
1.9.4: Proving Centers of Triangles

18. Key Concepts, continued
Theorem
Centroid Theorem
The centroid of a triangle is
the distance from each vertex to the midpoint of the opposite side.
The centroid of this triangle is at point X.
1.9.4: Proving Centers of Triangles

19. Key Concepts, continued
The centroid is always located inside the triangle.
The centroid is also called the center of gravity of a triangle because the triangle will always balance at this point.
Acute triangle
Obtuse triangle
Right triangle
1.9.4: Proving Centers of Triangles

20. Key Concepts, continued
Each point of concurrency discussed is considered a center of the triangle.
Each center serves its own purpose in design, planning, and construction.
Center of triangle
Intersection of…
Circumcenter
Perpendicular bisectors
Incenter
Angle bisectors
Orthocenter
Altitudes
Centroid
Medians
1.9.4: Proving Centers of Triangles

21. Common Errors/Misconceptions
not recognizing that the circumcenter and orthocenter are outside of obtuse triangles
incorrectly assuming that the perpendicular bisector of the side of a triangle will pass through the opposite vertex
interchanging circumcenter, incenter, orthocenter, and centroid
confusing medians with midsegments
misidentifying the height of the triangle
1.9.4: Proving Centers of Triangles

22. Guided Practice Example 3 has vertices A (–2, 4), B (5, 4), and
C (3, –2). Find the
equation of each median
of to verify that
(2, 2) is the centroid of
1.9.4: Proving Centers of Triangles

23. Guided Practice: Example 3, continued Identify known information.
has vertices A (–2,4), B (5, 4), and C (3, –2).
The centroid is X (2, 2).
The centroid of a triangle is the intersection of the medians of the triangle.
1.9.4: Proving Centers of Triangles

24. Guided Practice: Example 3, continued
Determine the midpoint of each side of the triangle.
Use the midpoint formula to find the midpoint of
Midpoint formula
Substitute (–2, 4) and (5, 4) for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

25. Guided Practice: Example 3, continued The midpoint of is .
Simplify.
1.9.4: Proving Centers of Triangles

26. Guided Practice: Example 3, continued
Use the midpoint formula to find the midpoint of
The midpoint of is (4, 1).
Midpoint formula
Substitute (5,4) and
(3, –2) for (x1, y1) and (x2, y2).
Simplify.
1.9.4: Proving Centers of Triangles

27. Guided Practice: Example 3, continued
Use the midpoint formula to find the midpoint of
Midpoint formula
Substitute (–2, 4) and
(3, –2) for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

28. Guided Practice: Example 3, continued The midpoint of is .
Simplify.
1.9.4: Proving Centers of Triangles

29. Guided Practice: Example 3, continued
1.9.4: Proving Centers of Triangles

30. Guided Practice: Example 3, continued
Determine the medians of the triangle.
Find the equation of , which is the line that passes through A and the midpoint of
Use the slope formula to calculate the slope of
Slope formula
Substitute (–2, 4) and (4, 1) for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

31. Guided Practice: Example 3, continued The slope of is
Simplify.
1.9.4: Proving Centers of Triangles

32. Guided Practice: Example 3, continued
Find the y-intercept of
The equation of that passes through A and the midpoint of is
Point-slope form of a line
Substitute (–2, 4) for
(x1, y1) and for m
Simplify
1.9.4: Proving Centers of Triangles

33. Guided Practice: Example 3, continued
Find the equation of , which is the line that passes through B and the midpoint of
Use the slope formula to calculate the slope of
Slope formula
Substitute (5,4) and
for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

34. Guided Practice: Example 3, continued The slope of is
Simplify.
1.9.4: Proving Centers of Triangles

35. Guided Practice: Example 3, continued
Find the y-intercept of
The equation of that passes through B and the midpoint of is
Point-slope form of a line
Substitute (5, 4) for
(x1, y1) and for m
Simplify
1.9.4: Proving Centers of Triangles

36. Guided Practice: Example 3, continued
Find the equation of , which is the line that passes through C and the midpoint of
Use the slope formula to calculate the slope of
Slope formula
Substitute (3, –2) and
for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

37. Guided Practice: Example 3, continued The slope of is
Simplify.
1.9.4: Proving Centers of Triangles

38. Guided Practice: Example 3, continued
Find the y-intercept of
The equation of that passes through C and the midpoint of is
Point-slope form of a line
Substitute (3, –2) for
(x1, y1) and –4 for m
Simplify
1.9.4: Proving Centers of Triangles

39. Guided Practice: Example 3, continued
Verify that X (2, 2) is the intersection of the three medians.
For (2, 2) to be the intersection of the three medians, the point must satisfy each of the equations:
1.9.4: Proving Centers of Triangles

40. Guided Practice: Example 3, continued
(2, 2) satisfies the equation of the median from A to the midpoint of
Equation of the median from A to the midpoint of
Substitute X (2, 2) for (x, y).
Simplify
1.9.4: Proving Centers of Triangles

41. Guided Practice: Example 3, continued
(2, 2) satisfies the equation of the median from B to the midpoint of
Equation of the median from B to the midpoint of
Substitute X (2, 2) for (x, y).
Simplify
1.9.4: Proving Centers of Triangles

42. Guided Practice: Example 3, continued
(2, 2) satisfies the equation of the median from C to the midpoint of
Equation of the median from C to the midpoint of
Substitute X (2, 2) for (x, y).
Simplify
1.9.4: Proving Centers of Triangles

43. ✔ Guided Practice: Example 3, continued State your conclusion.
X (2, 2) is the centroid of with vertices A (–2, 4), B (5, 4), and C (3, –2) because X satisfies each of the equations of the medians of the triangle. ✔
1.9.4: Proving Centers of Triangles

44. Guided Practice: Example 3, continued
1.9.4: Proving Centers of Triangles

45. Guided Practice Example 4 Using from Example 3, which has
vertices A (–2, 4),
B (5, 4), and C (3, –2),
verify that the centroid,
X (2, 2), is the
distance from each
vertex.
1.9.4: Proving Centers of Triangles

46. Guided Practice: Example 4, continued Identify the known information.
has vertices A (–2, 4), B (5, 4), and C (3, –2).
The centroid is X (2, 2).
The midpoints of are T , U (4, 1), and V
1.9.4: Proving Centers of Triangles

47. Guided Practice: Example 4, continued
Use the distance formula to show that point X (2, 2) is the distance from each vertex.
Use the distance formula to calculate the distance from A to U.
Distance formula
1.9.4: Proving Centers of Triangles

48. Guided Practice: Example 4, continued
The distance from A to U is units.
Substitute (–2, 4) and (4, 1) for (x1, y1) and (x2, y2).
Simplify.
1.9.4: Proving Centers of Triangles

49. Guided Practice: Example 4, continued
Calculate the distance from X to A.
Distance formula
Substitute (2, 2) and
(–2, 4) for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

50. Guided Practice: Example 4, continued
The distance from X to A is units.
Simplify.
1.9.4: Proving Centers of Triangles

51. Guided Practice: Example 4, continued X is the distance from A.
Centroid Theorem
Substitute the distances found for AU and XA.
Simplify
1.9.4: Proving Centers of Triangles

52. Guided Practice: Example 4, continued
Use the distance formula to calculate the distance from B to V.
Distance formula
Substitute (5, 4) and
for (x1, y1) and
(x2, y2).
Simplify.
1.9.4: Proving Centers of Triangles

53. Guided Practice: Example 4, continued
The distance from B to V is units.
1.9.4: Proving Centers of Triangles

54. Guided Practice: Example 4, continued
Calculate the distance from X to B.
Distance formula
Substitute (2, 2) and
(5,4) for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

55. Guided Practice: Example 4, continued
The distance from X to B is units.
Simplify.
1.9.4: Proving Centers of Triangles

56. Guided Practice: Example 4, continued X is the distance from B.
Centroid Theorem
Substitute the distances found for BV and XB.
Simplify
1.9.4: Proving Centers of Triangles

57. Guided Practice: Example 4, continued
Use the distance formula to calculate the distance from C to T.
Distance formula
Substitute (3, –2) and
for (x1, y1) and
(x2, y2).
Simplify.
1.9.4: Proving Centers of Triangles

58. Guided Practice: Example 4, continued
The distance from C to T is units.
1.9.4: Proving Centers of Triangles

59. Guided Practice: Example 4, continued
Calculate the distance from X to C.
Distance formula
Substitute (2, 2) and
(3, –2) for (x1, y1) and (x2, y2).
1.9.4: Proving Centers of Triangles

60. Guided Practice: Example 4, continued
The distance from X to C is units.
Simplify.
1.9.4: Proving Centers of Triangles

61. Guided Practice: Example 4, continued X is the distance from C.
Centroid Theorem
Substitute the distances found for BV and XB.
Simplify
1.9.4: Proving Centers of Triangles

62. ✔ Guided Practice: Example 4, continued
The centroid, X (2, 2), is the distance from each vertex. ✔
1.9.4: Proving Centers of Triangles

63. Guided Practice: Example 4, continued
1.9.4: Proving Centers of Triangles