This document discusses four types of triangle concurrency points: the circumcenter, incenter, centroid, and orthocenter. The circumcenter is the point equidistant from the vertices and allows circumscribing a circle around the triangle. The incenter is equidistant from the sides and allows inscribing a circle within the triangle. The centroid divides each median into ratios corresponding to the triangle's area ratios and allows the triangle to balance. The orthocenter's importance is said to be merely an interesting mathematical fact.

1. Congruent Triangles 4.52 Importance of concurrency

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

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

4. 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. 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 ? P X Y

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

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

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

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

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

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

22. 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 lines containing the altitudes are concurrent. The Orthocenter

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. Isn’t this FUN !!! Psych

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

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