1. Points of Concurrency Lessons 5.2-5.4
Prepared by Mrs. Pullo for her Geometry classes

2. Concurrent Lines: Circumcenter, Incenter, Centroid, Orthocenter
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. The perpendicular bisectors of the sides of a triangle are
Theorem 5-6
The perpendicular bisectors of the sides of a triangle are
concurrent at a point equidistant from the vertices called the
Circumcenter
Midpoint is found by adding the x coordinates and dividing by 2,
and y coordinates and dividing by 2.
To find the perpendicular
Line, must find slope and
take the negative reciprocal
Point of concurrency –
CIRCUMCENTER

4. The Circumcenter is equidistant from all three vertices.
Each line is perpendicular to the side and also bisects that side. The Perpendicular Bisector does not have to come from vertex!!

5. The Circumcenter gets its name from the fact
that it is the center of the circle that circumscribes
the triangle. Circumscribe means to be drawn
around by touching as many points as possible.

6. So, to find the center of a circle that will
circumscribe any given triangle, you need
to find the point of concurrency of the three
perpendicular bisectors of the triangle.
Sometimes this will be inside the triangle,
sometimes it will be on the triangle, and
sometimes it will be outside of the triangle!
Acute Right Obtuse

7. Theorem 5-7
The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides and are
Called the Angle bisectors.
Angle bisector
Incenter----

8. 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.

9. Median: A median of a triangle is the segment
That connects a vertex to the midpoint of the
Opposite side.

10. This point of concurrency of the Medians is another center of a triangle called the CENTROID.

11. From the Centroid to the vertex is 2/3 the length
of the median and from the Centroid to the edge is 1/3
the length of the median.

12. So, if you know the length of any median, you know
where the three medians are concurrent. It would be at the point that is 2/3 the length of the median
from the vertex it originated from.
Theorem 5-8

13. The Centroid is also the Center of Gravity of a triangle which means it is the point where a triangular shape will Balance.

14. Altitudes :
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.

15. Theorem 5-9
This point of concurrency of the altitudes
Of a triangle form another center of triangles.
This center is known as the Orthocenter.

16. In Conclusion: There are many centers of
Triangles. We have only looked at 4:
Circumcenter: Where the perpendicular bisectors meet
Incenter: Where the angle bisectors meet
Centroid: Where the medians meet
Orthocenter: Where the altitudes meet.

17. CONCURRENT LINES
Geometry Honors