1. Points of Concurrency
Objectives: To identify properties of perpendicular bisectors and angle bisectors

2. Points of concurrency
Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is called the point of concurrency.

3. Perpendicular Bisectors
Angle Bisectors
Medians
Altitudes

4. Perpendicular Bisectors
Angle Bisectors
Medians
Altitudes
Peanut Butter Cookies
Are Best In
Milk Chocolate
And Ovaltine
Perpendicular Bisectors - Circumcenter
Angle Bisector - Incenter
Medians - Centroid
Altitude - Orthocenter

5. Perpendicular Bisectors
circumcenter
vertex
Construct the perpendicular bisectors of each side of the triangle.
**NOT always inside the triangle. **

6. Perpendicular Bisectors
Construct the perpendicular bisectors of each side of the triangle.
Perpendicular Bisectors
circumcenter
vertex

7. Construct the angle bisectors from each vertex of the triangle.
incenter
side
Construct the angle bisectors from each vertex of the triangle.
**ALWAYS inside the triangle. **
The incenter is the center of the triangle's incircle, the largest circle that will fit inside the triangle and touch all three sides

8. Construct the angle bisectors from each vertex of the triangle.
incenter
side
Construct the angle bisectors from each vertex of the triangle.

9. Example #1 The perpendicular bisectors of ∆ABC meet at point D.
a. Find DB
Find AE
c. Find ED (Hint: Use the Pythagorean Theorem.) Write your answer in simplified radical form.

10. Example #2 R is the circumcenter of ∆OPQ. OS = 10,
QR = 12, and PQ = 22.
Find OP
b. Find RP
c. Find OR
d. Find TP
e. Find RT

11. Example #3
Your family is considering moving to a new home. The diagram shows the locations of where your parents work and where you go to school. The locations form a triangle.
In this diagram, how could you find a point that is equidistant from each location? Explain your answer.

12. homework 1. Centers of Triangles Worksheet
**Be prepared for a 10 minute Pop Quiz on today’s lesson next time. [10 points] ** It will be on circumcenter and incenter of a triangle.

13. Construct the median of each side of the triangle.
“center of gravity”
**Always INSIDE the triangle. **
Medians
The centroid is exactly two-thirds the way along each median.
Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.
median
vertex
midpoint

14. Construct the median of each side of the triangle.
“center of gravity”
Medians
Steps to find the centroid:
1. Find the midpoint of each side of the triangle.
2. Draw a segment joining the vertex of the triangle to the midpoint of the opposite
side.
3. The point of intersection of the three segments is the centroid.
median
vertex
midpoint

15. Construct the three altitudes of the triangle
perpendicular
Construct the three altitudes of the triangle
**NOT always inside the triangle. **
Obtuse
Acute
Right

16. Construct the three altitudes of the triangle
perpendicular
Construct the three altitudes of the triangle