1. Chapter 5.3 Concurrent Lines, Medians, and Altitudes

2. Vocabularies

3. Vocabularies
Concurrent (line) = three or more lines intersect in one point
Point of concurrency = the point the concurrent lines intersect
Example:

4. Vocabularies
Circumcenter of the triangle = The point of concurrency of the perpendicular bisectors of a triangle
Example:

5. Vocabularies
Circumscribed about = A circle is circumscribed about a polygon if the vertices of the polygon are on the circle. A polygon is circumscribed about a circle if all the sides of the polygon are tangent to the circle
Examples:

6. Vocabularies
Incenter of the triangle = The incenter of a triangle is the point of concurrency of the angle bisectors of the triangle
Example:

7. Vocabularies
Inscribed in = A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle. A polygon is inscribed in a circle if the vertices of the polygon are on the circle
Examples:

8. Vocabularies
Median of a triangle = a segment whose endpoints are a vertex and the midpoint of the opposite side
Examples:

9. Vocabularies
Centroid = the point of intersection of the medians of that triangle
Example:

10. Vocabularies
Altitude of a triangle = the perpendicular segment from a vertex to the line containing the opposite side. Can be inside or outside the triangle.
Examples:
Acute Triangle Right Triangle Obtuse Triangle

11. Vocabularies
Orthocenter of the triangle = the point of intersection of the lines containing the altitudes of the triangle.
Example:

12. Theorems

13. Theorem 5.6
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices
Example:

14. Theorem 5.7
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides
Example:

15. Theorem 5.8
The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side
Example:

16. Theorem 5.9
The lines that contain the altitudes of a triangle are concurrent

17. Classwork/Homework