1. 5-3
Concurrent Lines, Medians, Altitudes
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.

2. 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.
Point of concurrency

3. This point of concurrency has a special name.
It is known as the circumcenter of a triangle. The
circumcenter of a triangle is one of many different
“centers” of a triangle.
Circumcenter

4. The circumcenter is equidistant from all three
Vertices.

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.
Angle bisector

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. Theorem 5-8
The medians of a triangle are concurrent at a point
that is two thirds the distance from each
vertex to the midpoint of the opposite side.

11. Theorem 5-8
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.

12. For Example:
If you know a median of a triangle
Is 12cm, you could determine the point of concurrency
Of all three medians (2/3 of 12) or 8cm from the vertex. 8

13. This point of concurrency of the
Medians is another center
Of a triangle.
It is known as the Centroid

14. This Centroid
Is also the center of Gravity
Of a triangle which means it is the
Point where a triangular shape will
Balance.

15. 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.

16. This point of concurrency of the altitudes
Of a triangle form another center of triangles.
This center is known as the
Orthocenter.

17. Theorem 5-9
The lines that contain the altitudes of a triangle are
Concurrent.

18. 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.