1. Y. Davis Geometry Notes
Chapter 5

2. Perpendicular Bisectors
A segment, ray, line or plane that is
perpendicular to a segment at its midpoint.

3. Theorem 5.1 Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector
of a segment, then it is equidistant from the
endpoints of the segment.

4. Theorem 5.2 Converse of Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of
a segment, then it lies on the perpendicular
bisector of the segment.

5. Concurrent Lines
3 or more lines that intersect at the same
point.

6. Point of Concurrency
The point where 3 or more lines intersect.

7. Circumcenter The point of concurrency for perpendicular
bisectors of a triangle.

8. Theorem 5.3 Circumcenter Theorem
The perpendicular bisectors intersect at a
point called the circumcenter, that is
equidistant from the vertices of the triangle.

9. Theorem 5.4 Angle Bisector Theorem
If a point lies on the angle bisector, then it is
equidistant from the sides of the angle.

10. Theorem 5.5 Converse of the Angle Bisector Theorem
If a point is equidistant from the sides of the
angle, then it lies on the angles bisector.

11. Incenter The point of concurrency for the angle
bisectors of a triangle.

12. Therorem 5.6 Incenter Theorem
The angle bisector of a triangle intersect at a
point called the incenter, that is equidistant
from the sides of the triangle.

13. Medians
Is a segment with endpoints on the vertex of a triangle and the midpoint of the opposite side.

14. Centroid
The point of concurrency for the medians of a triangle.

15. Theorem 5.7 Centroid Theorem
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.

16. Altitude
Is a perpendicular line from the vertex to the line containing the opposite side.

17. Orthocenter
The point of concurrency for the altitudes of a triangle.

18. Equidistant from sides
Concurrent Lines
Point of Currency
Facts
Locations
(Triangles)
Perpendicular bisectors
Circumcenter
Equidistant from vertices
Acute—Inside
Right—Inside
Obtuse--Inside
Angle Bisectors
Incenter
Equidistant from sides
Right—On(hypotenuse)
Obtuse--Outside
Medians
Centroid
2/3 the length from vertex to midpoint
Altitudes
Orthocenter
_____
Right—On (Right angle)

19. Inequality
For all real numbers, a>b, if and only if there is a positive number c, so that a=b+c

20. Theorem 5.8 Exterior Angle inequality
The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

21. Angle-Side Relationships in Triangles
Theorem 5.9—In a triangle, the longest side is opposite the largest angle. The shortest side is opposite of the smallest angle.
Theorem 5.10—In a triangle, the largest angle is opposite the longest side. The smallest angle is opposite the shortest side.

22. Indirect reasoning
Assume that the conclusion is false and then show that the assumption led to a condradiction.

23. Indirect Proof Proof by contradiction.
Assume the conclusion is false, by assuming the that the opposite is true.
Use logical reasoning to show that this leads to a contradiction of the hypothesis, definition, postulate, theorem, or corollary.
Since the assumption leads to a contradiction, then the original conclusion must be true.

24. Theorem 5.11 Triangle Inequality Theorem
The sum of any two sides of a triangle must be greater than the length of the third side.

25. Theorem Hinge Theorem
If 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angle of the 1st triangle is larger than the included angle of the 2nd triangle, then the 3rd side of the 1st triangle is longer than the 3rd side of the 2nd triangle.

26. Theorem 5.14 Converse of the Hinge Theorem
If 2 sides of one triangle are congruent to 2 sides of another triangle, and the 3rd side of the 1st triangle is longer than the 3rd side of the 2nd triangle, then the included angle of the 1st triangle is larger than the include angle of the 2nd triangle.