1. LESSON FIFTEEN: TRIANGLES IN TRAINING

2. MORE TRIANGLE PROPERTIES In the last lesson, we discussed perpendicular bisectors and how they intersect to create a circumcenter. We also discussed angle bisectors and how they intersect to create incenters.

3. MORE TRIANGLE PROPERTIES Today, we’ll discuss two new points of concurrence and the lines that intersect to create them.

4. MORE TRIANGLE PROPERTIES The first type of line we will discuss is the altitude and you’ll find that this is actually a term you already know. The altitude of a triangle is the segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. So it’s the same as the height.

5. MORE TRIANGLE PROPERTIES Quickly draw the altitudes for each angle of the triangle below. What do you notice about the lines?

6. MORE TRIANGLE PROPERTIES They are concurrent! The point at which they meet is called the orthocenter.

7. MORE TRIANGLE PROPERTIES The orthocenter can be inside, outside or directly on the triangles

8. MORE TRIANGLE PROPERTIES The orthocenter is the only point of our four, that has no special properties. The only thing about it to remember is the information from the previous slide.

9. MORE TRIANGLE PROPERTIES The next concurrent lines we will discuss are called medians. A median is a segment with the endpoints being a vertex of a triangle and the midpoint of the opposite side.

10. MORE TRIANGLE PROPERTIES The medians of a triangle are also concurrent. The point at which they meet is called the centroid.

11. MORE TRIANGLE PROPERTIES The centroid of any triangle (like the incenter) is ALWAYS on the inside of the triangle.

12. MORE TRIANGLE PROPERTIES The centroid theorem states that the medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.

13. MORE TRIANGLE PROPERTIES Finally, we can construct the Euler Line. The Euler Line is the line that runs between the orthocenter, circumcenter, and centroid. Centroid Orthocenter Circumcenter

14. MORE TRIANGLE PROPERTIES Now we can construct the Euler Line. Centroid Orthocenter Circumcenter

15. MORE TRIANGLE PROPERTIES In an equilateral triangle the orthocenter, circumcenter and centroid will be the same! Thus, technically there is no Euler Line for them.