1. Triangle Centers Students will identify triangle centers and explain properties associated with them.

2. There are many types of triangle centers. Below are four of the most common Incenter Located at intersection of the angle bisectors. See Triangle incenter definitionTriangle incenter definition Circumcenter Located at intersection of the perpendicular bisectors of the sides. See Triangle circumcenter definitionTriangle circumcenter definition Centroid Located at intersection of medians. See Centroid of a triangleCentroid of a triangle Orthocenter Located at intersection of the altitudes of the triangle. See Orthocenter of a triangleOrthocenter of a triangle

3. In the case of an equilateral triangle, all four of the above centers occur at the same pointequilateral triangle

4. Euler’s Line Euler’s Line – demonstrationdemonstration How can this be helpful in determining facts about a triangle?

5. Deserted Island Task http://illuminations.nctm.org/LessonDetail.as px?id=L660 http://illuminations.nctm.org/LessonDetail.as px?id=L660 Applet for polygon distances - Applet for polygon distances

6. Why does it make sense that the sum of the distances from an interior point to the sides of a regular polygon with an even number of sides is a multiple of the height of the polygon?

7. [Since the regular polygons with even number of sides will have pairs of sides parallel, the combined distance for each these pair will equal the height of the polygon. Also, the number of pairs will be one ‑ the number of sides of the polygon. So, the sum will always be n/2 times the height of the polygon. The square portion of the Triangle Island applet can be used to show this for the specific case of a four ‑ sided polygon, which may help students see that it would be true for other even ‑ sided polygons.] Triangle Island

8. For the case of the regular triangle, the sum of the distances from any interior point to the three sides is equal to the height of the triangle. Prove this result. [Subdivide the triangle into three smaller triangles, as shown below. Because the triangle is equilateral, AB = BC = CA.

9. The area of the yellow triangle is ½(AB)(HR); the area of the green triangle is ½(BC)(HQ); and, the area of the red triangle is ½(CA)(HP). Consequently, the area of the triangle is ½(AB)(HR) + ½(BC)(HQ) + ½(CA)(HP). But because the sides are equal, this can be rewritten as ½(AB)(HR + HQ + HP). In addition, the area of the triangle is equal to ½(height)(base) = ½(h)(AB). Setting these two results equal gives: ½(AB)(HR + HQ + HP) = ½(h)(AB) Dividing both sides by ½(AB) gives HR + HQ + HP = h, so the sum of the distances to each side is equal to the height of the triangle.

10. What does this observation tell us about the relationship between the height of an equilateral triangle and the diameter of its circumscribed circle?