Points of Concurrency:
Name:
Incenter, Orthocenter, Circumcenter, Centroid
Sometimes when you don’t know much of anything about a topic, you find all the information you can. You
dissect it, make sense of the parts, and put it all back together again. That’s what you’ll need to do here.
Day
Day 1
Objective
 Conduct research
 Construct points of
concurrency use compass
and straightedge.
Homework
Construct an organize summary of
your findings. There should be 4
facts per point including an
illustration. (12 hmwk pts)
Day 2

Solve problems involving
points of concurrency.
Worksheet Day 2
Day 3

Solve problems using
coordinate geometry.
Worksheet Day 3
Day 4



Quest (40 test pts)
Due: Find Euler’s Line (10 project pts)
Due: The completion of the note-taking guide. (20 hwk pts)
1 |P o i n t s o f C o n c u r r e n c y
Day 1 – Conduct the Research; Construct the points.
Use the following chart to organize your findings.
Incenter
Orthocenter
2 |P o i n t s o f C o n c u r r e n c y
Circumcenter
Centroid
Constructing Points of Concurrency
(points of intersection)
Constructing a Midpoint.
Constructing an Altitude through a point.
Constructing a Perpendicular Bisector.
Bisecting an Angle
3 |P o i n t s o f C o n c u r r e n c y
A. In the triangle below, construct a median AX.
A
B
A
C
B. In the triangle below, construct an altitude AX.
B
C. In the triangle below, construct an angle bisector AX.
B
C
C
A
D. In the triangle below, construct a perpendicular bisector CX.
B
C
4 |P o i n t s o f C o n c u r r e n c y
A
Be sure to mark diagrams appropriately, no assumptions are to be made.
1. Construct the centroid in the triangle below.
A
B
C
2. Construct the circumcenter in the triangle below.
A
B
C
3. Construct the orthocenter in the triangle below.
A
B
C
4. Construct the incenter in the triangle below.
A
B
5 |P o i n t s o f C o n c u r r e n c y
C
Day 2 - Solve problems involving points of concurrency.
Solve each problem below in one word.
1. In what kind of triangle is the orthocenter a vertex of the triangle?
2. Which point of concurrency is formed by the intersection of the medians?
3. For which point of concurrency are the segments of intersection not drawn from the vertices of the
triangles?
4. If a triangle is cut from cardboard and the circumcenter, the orthocenter, the centroid, and the
incenter are located, upon which point could the triangle be balanced?
5. Every triangle has a circumcenter, an orthocenter, a centroid, and an incenter. Which of the four
points will always lie in the interior of the triangle?
6. The first-aid center of Mt. Thermopolis State Park needs to be at a point that is equidistant from
three bike paths that intersect to form a triangle. What is the point called that will enable the
emergency medical personnel to be able to get to any one of the paths by the shortest route
possible?
7. A stained-glass artist whished to circumscribe a circle about a triangle in her latest abstract design.
Which point of concurrency does she need to locate?
8. Rosita wants to install a circular sink in her new triangular countertop. She wants to choose the
largest sink that will fit. Which point of concurrency must she locate?
6 |P o i n t s o f C o n c u r r e n c y
9. Given: ABC, with medians AM ,BN ,and CP . If AM = 9, find AT.
A
P
N
T
B
M
C
10. Given: ABC, with medians AM ,BN ,and CP . If TN = 5, find BN.
A
P
N
T
B
M
C
11. Given: ABC, with medians AM ,BN ,and CP . If TC = 6, find PT.
A
P
N
T
B
M
C
12. Given: ABC, with medians AM ,BN ,and CP . If BN = 12 , find TN.
A
P
T
N
B
M
7 |P o i n t s o f C o n c u r r e n c y
C
13. Point M is the centroid. Find the missing information:
a. CM = 16
C
b. MO = 10
S
c. TS = 21
M
O
d. AM = ___
A
e. SM = ___
U
T
14. Given: PLO with centroid V. VT = 6, AT = 9, OT = 18. Find PA.
P
V
L
T
A
O
15. Given: PLO with centroid V. VT = 6, AT = 9, OT = 18. Find the area POT.
P
V
L
8 |P o i n t s o f C o n c u r r e n c y
A
T
O
Day 3 - Solve problems using coordinate geometry.
CENTROIDS
ORTHOCENTER
9 |P o i n t s o f C o n c u r r e n c y
ORTHOCENTER
CIRCUMCENTER
10 |P o i n t s o f C o n c u r r e n c y