1. Congruent Triangles
4.52 Importance of concurrency

2. Triangle Points of Concurrency#1
Perpendicular bisectors
and
the circumcenter.

3. Circumcenter
Before we can talk about the circumcenter’s importance, we need some review on perpendicular bisectors.

4. perpendicular bisector is equidistant from A and B.
Review of Perpendicular Bisector Properties.
Every point on the
perpendicular bisector
is equidistant from A and B.

5. Therefore, the circumcenter is equidistant from
each vertex.
Why ?

6. Since C is on each perpendicular bisector it is equidistant from each segment’s endpoint.
Therefore…

7. Ergo the term… Circumcenter.
The equal distances are radii for a circle that is …
written around (circumscribed) the circle.
Ergo the term… Circumcenter.

8. Why is this important?
If A, B, and T are three cities, Point C is the ideal place to build a communication tower to broadcast to each city.

9. Triangle Points of Concurrency#2
Angle bisectors
and
the Incenter.

11. From any general point on the angle bisector, the perpendicular distance to either side is the same.
Why ? Y X P

12. The top angles are congruent by definition of angle bisector.
There are two right angles are congruent by def. of perpendicular. Y
by the reflexive property of = X ?
By AAS P ?
By CPCTC

13. Let’s see. Why is this important ? Each new point creates
2 congruent triangles by AAS.
Why is this important ?
Let’s see.

14. Since the incenter P is on each angle bisector….

15. P Why is this important? Point P is equidistant from each side.
Remember, the distance from a point to a line is the brown perpendicular segment. P
Why is this
important?
Because these equal distances are radial distances.

16. Radial distances refer to a circle.
The incenter generates an inscribed circle. P

17. Triangle Points of Concurrency#3
Medians
and
the Centroid.

18. A median is a segment connecting the vertex to the midpoint of a side of a triangle.

19. The 3 medians meet at the centroid – point P.

20. Each little triangle is unique, yet they all have something in common.
What is it ?

21. the triangle will balance on the centroid.
Since the areas of the little triangles around the centroid E are the same, …
the triangle will balance on the centroid.

22. Calder’s Mobile at the East Wing of the National Gallery of Art
Mobiles balance objects in an artistic form.
Calder’s Mobile at the East Wing of the National Gallery of Art
In Washington DC.

23. Triangles balanced at their centroids.

24. The birds are tied to their centers of balance
or centroids.

25. Triangle Points of Concurrency#4
Altitudes
and
the Orthocenter.

26. Acute Triangle
The point of concurrency is called…
The Orthocenter

27. Right Triangle
The point of concurrency is called…
The Orthocenter

28. Obtuse Triangle
Notice that although the altitudes are not concurrent…

29. Obtuse Triangle The Orthocenter
The lines containing the altitudes are concurrent.

30. So what do you think is the importance or practical application of the orthocenter is?
Nothing !!!
It is just an interesting fact that mathematicians have discovered.
Psych
Isn’t this FUN !!!

31. Summary None Medians Orthocenter Altitudes Circumcenter Incenter
Line Type
Concurrency
Point
Importance
Equidistant from the vertices
Circumcenter
Circumscribed Circle
Equidistant from the sides
Incenter
Inscribed Circle
Medians
Centroid
Center of Balance
None
Altitudes
Orthocenter

32. C’est fini. Good day and good luck. A Senior Citizen Production
That’s all folks.
A Senior Citizen Production