1. POINTS OF CONCURRENCY
In this lesson we will define what a point of concurrency is.
Then we will look at 4 points of concurrency in triangles.
As you go through the powerpoint, you will complete your
notesheet.
You will need to be able to define the 4 points of
concurrency and identify them in a picture. You will also
use the definition to identify relationships between
Segments and angles to solve problems.

2. POINTS OF CONCURRENCY
When two lines intersect at one point, we say that the lines
are intersecting. The point at which they intersect is
the point of intersection.
(nothing new right?)
Well, if three or more lines intersect, we say that the lines
are concurrent. The point at which these lines intersect
Is called the point of concurrency.

3. Perpendicular bisectors=Circumcenter
The perpendicular bisectors of the sides of a triangle are
concurrent at a point equidistant from the vertices.
Point of concurrency

4. Perpendicular bisectors=Circumcenter
This point of concurrency has a special name. It is known as the circumcenter of a triangle.
Circumcenter

5. Perpendicular bisectors=Circumcenter
The circumcenter is equidistant from all three vertices.

6. Perpendicular bisectors=Circumcenter
The circumcenter gets its name because it is the center of the circle that circumscribes the triangle. Circumscribe means to be drawn around by touching as many points as possible.

7. Perpendicular bisectors=Circumcenter
Sometimes the circumcenter will be inside the triangle,
sometimes it will be on the triangle, and
sometimes it will be outside of the triangle!
Acute Right Obtuse

8. Angle bisectors=Incenter
The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides.
Angle bisector

9. Angle bisectors=Incenter
The point of concurrency of the three angle bisectors is another center of a triangle known as the Incenter. It is equidistant from the sides of the triangle, and gets its name from the fact that it is the center of the circle that is inscribed within the circle.
Incenter

10. Medians=Centroid
Median: A median of a triangle is the segment That connects a vertex to the midpoint of the opposite side.

11. Medians=Centroid
This point of concurrency of the medians is another center of a triangle. It is known as the Centroid.
Centroid

12. Medians=Centroid
This Centroid is also the Center of Gravity of a triangle which means it is the point where a triangular shape will
balance.

13. Centroid Centroid Centroid Theorem
The point of concurrency of the medians of a triangle.
Centroid Theorem
The medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.

14. Centroid… PK = 5 Find AP 10 x 2x BP = 12 Find PL. 6 JC = 15
PK + AP = AK
JC = 15
Find JP. 5
PK + 2(PK) = AK

15. In triangle ABC, Q is the centroid and BE = 9
Find BQ
Find QE
BQ = 2(QE)
= 3 OR
6 = 2(QE)
3 = QE

16. In triangle ABC, Q is the centroid and FC = 15
Find FQ
Find QC
= 5
QC = 2(FQ)
QC = 10
QC = 2(5)

17. Altitudes=Orthocenter
Altitudes of a triangle are the perpendicular
segments from the vertices to the line containing
the opposite side.
Unlike medians, and angle bisectors that are
always inside a triangle, altitudes can be inside,
on or outside the triangle.

18. Altitudes=Orthocenter
This point of concurrency of the altitudes of a triangle form another center of triangles. This center is known as the Orthocenter.
The Orthocenter of a triangle can be inside
on or outside of the triangle.

19. Find the orthocenter
The vertices of triangle FGH are F(-2, 4), G(4,4), and H(1, -2). Find the coordinates of the orthocenter of triangle FGH.
Graph the points.
Cont…

20. Find an equation from F to GH.
Slope of GH.
m = 2
New equation is perpendicular to segment GH.
Point F (-2, 4) m = -1/2
y = mx + b
3 = b
Cont…

21. Find an equation from G to FH.
Slope of segment FH
m = -2
New equation is perpendicular to segment FH.
Point G (4, 4) m = 1/2
2 = b
Cont…

22. The orthocenter can be found at the intersection of our 2 new equations.
How can we find the orthocenter?
If the orthocenter lies on an exact point of the graph
use the graph to name. If it does not lie on a point
use systems of equations to find the orthocenter.
System of equations:
Cont…

23. By substitution.
Orthocenter
(1, 2.5)
1 = x
y = 2.5

24. POINTS OF CONCURRENCY
There are many Points of Concurrency in Triangle. We
have only looked at four:
Circumcenter: Where the perpendicular bisectors meet
Incenter: Where the angle bisectors meet
Centroid: Where the medians meet
Orthocenter: Where the altitudes meet.

25. Practice Problems On the half sheet of paper that you were given
complete the following problems.
1A-C Identify the point of concurrency that is shown
in the triangle B C A

26. Practice Problems
Solve for x, y in each triangle using the given information
2. Point G is a centroid
AC = 24, AF=15,
AE= 3x-6, BF = 3y
3. Point H is an orthocenter
<ABE = 25, <BFC = 8x + 10,
<BAE= 6y+5

27. Practice Problems 4. Point W is both a centroid and an orthocenter.
Why is it a circumcenter also?
5. If you draw two medians, is that enough to determine where the centroid of a triangle is? Why/Why not?
6. Do the following matching.
Circumcenter W. Angle bisectors
Incenter X. Medians
Orthocenter Y. Perpendicular bisectors
Centroid Z. Altitudes