Concurrent Lines within a Triangle

1. Concurrent Lines within a Triangle How can you best describe the different lines within a triangle?

2. Vocabulary Concurrent lines – Three or more lines that intersect at a common point. Point of concurrency – The point where the lines intersect. Equidistant – Equal distance.

3. Special Lines of a Triangle Angle Bisector Lines A line drawn from the vertex of a triangle to the opposite side of a triangle. The angle bisector line bisects the angle from the vertex. There are three angle bisector lines.

4. Special Lines of a Triangle • The point of concurrency is called the incenter. • The incenter is equidistant to the sidesand the line from them to the sides is perpendicular.

5. Special Lines of a Triangle Perpendicular Bisector Lines A line drawn from the side of one triangle to the opposite side. It bisects the side it’s drawn from and creates a 90 degree angle with that side. There are three perpendicular bisector lines for each triangle.

6. Special Lines of a Triangle • The point of concurrency is called the circumcenter. • The circumcenter is equidistant to the vertices of a triangle. • It’s possible for the circumcenter to be located on or outside of the triangle.

7. Special Lines of a Triangle Some perpendicular bisector lines can be drawn from the vertex of a triangle. Only true with equilateral triangles and isosceles triangles.

8. Special Lines of a Triangle Altitudes A line drawn from the vertex to the opposite side. The altitude creates a right angle with the side. Some altitudes can be drawn outside of a triangle. Only obtuse triangles. There are three altitude lines.

9. Special Lines of a Triangle • The point of concurrency is called the orthocenter.

10. Special Lines of a Triangle Median A line inside the triangle drawn from the vertex to the opposite side. The median bisects the opposite side. There are three median lines.

11. Special Lines of a Triangle The point of concurrency is called the centroid. The centroid is located 2/3 of the way from the vertex. The portion of the median from the centroid to the vertex is double the distance of the centroid the opposite side.

12. Vocabulary • Equidistant – Equal distance. • Incenter – the point of concurrency of the angle bisectors lines. • Orthocenter – the point of concurrency of the altitudes.

13. Vocabulary • Centroid – the point of concurrency of the medians • Circumcenter – the point of concurrency of the perpendicular bisectors

14. Special Lines The circumcenter is equidistant from the vertices of a triangle. The circumcenter will meet on the outside of a triangle in an obtuse triangle. The circumcenter will meet on the triangle in a right triangle.

15. Special Lines The incenter is equidistant from each side of the triangle. The centroid is two-thirds the distance from each vertex to the midpoint of the opposite side.