1. Triangle Centers - Overview Thousands of years ago, when the Greek philosophers were laying the first foundations of geometry, someone was experimenting with triangles. They bisected two of the angles and noticed that the angle bisectors crossed. They drew the third bisector and surprised to find that it too went through the same point. They must have thought this was just a coincidence. But when they drew any triangle they discovered that the angle bisectors always intersect at a single point! This must be the 'center' of the triangle. Or so they thought.angle bisectors

2. Triangle Centers - Overview After some experimenting they found other surprising things. For example the altitudes of a triangle also pass through a single point (the orthocenter). But not the same point as before. Another center! Then they found that the medians pass through yet another single point. Unlike, say a circle, the triangle obviously has more than one 'center'.altitudesmedians The points where these various lines cross are called the triangle's points of concurrency.points of concurrency

3. Sections 5.2-5.4 Points of Concurrency There are many points of concurrency in a triangle, however, we will only Learn about four of them:the incenter, the circumcenter, the orthocenter, And the centroid.

4. Concurrent When three or more lines intersect in one point they are concurrent

5. Point of Concurrency The point at which three or more lines intersect is the point of concurrency Point of concurrency

6. The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

7. Circumcenter The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle. Circumcenter

8. The circle is circumscribed about the triangle. The circumcenter is the center of the circumscribed circle.

9. The circle is circumscribed about the triangle. To circumscribe a circle about a triangle, place your compass point on the circumcenter and the radius should extend to any of the three vertices. Now, construct your circle.

10. The towns of Adamsville, Brooksville, and Cartersville want to build a library that is equidistant from the three towns.Trace the diagram and show where they should build the library? Draw segments connecting the towns. Build the library at the intersection point of the perpendicular bisectors of the segments. The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

11. The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

12. Incenter The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. incenter

13. The circle is inscribed in the triangle The center of the inscribed circle is the incenter of the circle. The center is equally Distant from the three sides. (Remember: it is the perpendicular distance from the Center that determines the point at which the circle touches the side of the triangle, so construct a perpendicular from the incenter to a side of the triangle and use this as the radius of your inscribed circle.

14. Example of the incenter Question about Pools: The Jacksons want to install the largest possible circular pool in their triangular backyard.Where would the largest possible pool be located? Answer:Locate the center of the pool at the point of concurrency of the angle bisectors.This point is equidistant from the sides of the yard. If you choose any other point as the center of the pool, it will be closer to at least one of the sides of the yard,and the pool will be smaller.

15. A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side. Midpoint Median

16. Centroid In a triangle, the point of concurrency of the medians is the centroid. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance. There is a special property of a centroid

17. Look closely: What is the relationship between the centroid to the midpoint and the centroid to the vertex?

18. Altitudes of triangles An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle.

19. The lines containing the altitudes of a triangle are concurrent at the orthocenter of the triangle. orthocenter

20. FYI: Useless Information The intersection of the three altitudes of a triangle is called the orthocenter. The name was invented by Besant and Ferrers in 1865 while walking on a road leading out of Cambridge, England in the direction of London (Satterly 1962).altitudestriangle

21. Try this: Find the center of the circle that you can circumscribe about a right triangle with these vertices:

22. You need an equation for the perpendicular bisectors to circumscribe a circle about a triangle You only need to find the equation for two sides. I would use sides AB and BC since they are horizontal and vertical. Much easier. First, find the midpoints of those sides. Now find the equation of the perpendicular bisectors. Then find the intersection point of the two equations. Or…since this is a right triangle, we should know that the circumcenter is at the midpoint of the hypotenuse. Much easier to find!

24. 5) Altitude 6)Median 7) Perpendicular Bisector 8) Angle Bisector

25. Euler Line The incenter is NOT ALWAYS on the Euler line. The other three points ARE always on the Euler line.

26. Euler Line The incenter will be on the Euler line when the triangle is isosceles. The other three points ARE always on the Euler line. In an equilateral triangle, the 4 points will always coincide.

27. On the Euler Line above, point O is the circumcenter, point G is the centroid, point F is the center of the nine point circle and point H is the orthocenter. GO is ½ of HG OG is 1/3 of HO OF is ½ of HO FG is 1/6 of HO

28. Know all vocabulary, know facts about an angle bisector and a perpendicular bisector. Know how to construct each of these points of concurrency. Know how to inscribe and circumscribe a circle in relation to a triangle.