Points of Concurrency in Triangles
In this investigation, you will review the following:
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What are the centroid, the incenter, the circumcenter, and the orthocenter of a
triangle?
How do the points of concurrency in triangles solve problems?
SET UP:
Print four copies of the triangle below. Number the triangles 1, 2, 3, and 4.
Construct :
a. On triangle 1:
vertex.
b. On triangle 2:
c. On triangle 3:
d. On triangle 4:
side.
the midpoint of each side. Connect the midpoint to the opposite
the angle bisector of each angle.
the perpendicular bisector of each side.
the altitude from each vertex to the line containing the opposite
A
B
C
What do you notice about each one of these constructions?
Hopefully in each case you have noticed that there is a point of intersection (a point of
concurrency) that occurs. This is an important property of constructions within triangles
and one that solves many problems in mathematics.
Investigation:
In the Launch you constructed four different points of concurrency.
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The point of concurrency of the medians of a triangle is called the centroid. The
centroid is sometimes referred to as the “weighted center” or balancing point of a
triangle.
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The point of concurrency of the angle bisectors of a triangle is called the incenter.
This point is the center of a circle that inscribes the triangle.

The point of concurrency of the perpendicular bisectors of a triangle is called the
circumcenter. This point is the center of a circle that circumscribes the triangle.
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The point of concurrency of the altitudes of a triangle is called the orthocenter.
Draw other triangles (compare acute, obtuse, right triangles) and construct the centroid,
incenter, circumcenter, and orthocenter of each triangle. Answer each of the following
questions and explain how you know.
1. Does the centroid always lie inside the triangle?
Yes the centroid always lies inside the triangle because it is the intersection point of the
medians i.e. the lines joining the vertex to the mid-point of opposite side.
2. Does the incenter always lie inside the triangle?
Yes, the incenter always lies inside the triangle because it is the intersection point of the
angle bisectors which meet inside the triangle only.
3. Does the circumcenter always lie inside the triangle?
No, the circumcenter does not always lie inside the triangle. Circumcenters is the points at
which the perpendicular bisectors of the three sides meet. Like in a right triangle, the
circumcenter lies at the mid-point of the hypotenuse. In other triangles, it can even lie outside
and on the triangle.
4. Does the orthocenter always lie inside the triangle?
No, the orthocenter does not always lie inside the circle. Like in a right triangle, it lies on the
vertex where the right angle is. For obtuse angles triangles, it lies outside the triangle.
5. Can these points ever coincide?
Yes, for an equilateral triangle, all the four coincide and so the same point represents all the
four.
Conclusions:
What is the difference between the centroid, circumcenter, incenter, orthocenter
of a triangle?
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Circumcenter - intersection of the sides' perpendicular bisectors
Incenter - intersection of 3 angle bisectors
Centroid - intersection of 3 medians (middle of sides)
Orthocenter - intersection of 3 altitudes