1. 5.3 Concurrent Lines, Medians, and Altitudes
Chapter 5
Relationships Within Triangles

2. 5.3 Concurrent Lines, Medians, and Altitudes
Concurrent: When three or more lines intersect in one point
Point of concurrency: The point where three or more lines intersect

3. 5.3 Concurrent Lines, Medians, and Altitudes
Theorem 5-6
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices
Theorem 5-7
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides

4. Circumcenter
Circumcenter of the triangle: The point of concurrency of the perpendicular bisectors
Points Q, R, and S are equidistant from C, the circumcenter
The circle is circumscribed about the triangle S C R Q
Perpendicular Bisectors

5. Incenter
The incenter of the triangle is the point of concurrency of the angle bisectors
Points X, Y, and Z are equidistant from I, the incenter.
The circle is inscribed in the triangle T Y
Angle Bisector I X V U Z

6. Median of a Triangle
The median of a triangle is a segment that goes from the vertex to the midpoint of the opposite side.

7. Theorem 5-8
The medians of a triangle are concurrent at a point that is two third the distance from each vertex to the midpoint of the opposite side 8 3 6 4

8. Centroid
The point of concurrency of the medians is the Centroid

9. Altitude of a Triangle
Altitude: perpendicular segment from a vertex to the line containing the opposite side.
* The altitude can be inside the triangle,
outside the triangle, or a leg of the triangle

10. Orthocenter of the Triangle
The lines containing the altitudes of a triangle are concurrent at the orthocenter.

11. Identifying Medians and AltitudesW V T U

12. Practice
Pg and 19-22