1. Proving Centers of Triangles
Adapted from Walch Education

2. 1.9.4: Proving Centers of Triangles
Circumcenter
The perpendicular bisector is the line that is constructed through the midpoint of a segment.
The three perpendicular bisectors of a triangle are concurrent, or intersect at one point.
This point of concurrency is called the circumcenter of the triangle.
The circumcenter of a triangle is equidistant, or the same distance, from the vertices of the triangle. This is known as the Circumcenter Theorem.
1.9.4: Proving Centers of Triangles

3. 1.9.4: Proving Centers of Triangles
Theorem
Circumcenter Theorem
The circumcenter of a triangle is equidistant from the vertices of a triangle.
The circumcenter of this triangle is at X.
1.9.4: Proving Centers of Triangles

4. Circumcenter, continued
The circumcenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle.
The circumcenter is inside acute triangles, outside obtuse triangles, and on the midpoint of the hypotenuse of right triangles.
1.9.4: Proving Centers of Triangles

5. Circumcenter, continued
Acute triangle
Obtuse triangle
Right triangle
X is inside the triangle.
X is outside the triangle.
X is on the midpoint of the hypotenuse.
1.9.4: Proving Centers of Triangles

6. Circumcenter, continued
The circumcenter of a triangle is also the center of the circle that connects each of the vertices of a triangle. This is known as the circle that circumscribes the triangle.
1.9.4: Proving Centers of Triangles

7. 1.9.4: Proving Centers of Triangles
Incenter
The angle bisectors of a triangle are rays that cut the measure of each vertex in half.
The three angle bisectors of a triangle are also concurrent.
This point of concurrency is called the incenter of the triangle.
The incenter of a triangle is equidistant from the sides of the triangle. This is known as the Incenter Theorem.
1.9.4: Proving Centers of Triangles

8. 1.9.4: Proving Centers of Triangles
Theorem
Incenter Theorem
The incenter of a triangle is equidistant from the sides of a triangle.
The incenter of this triangle is at X.
1.9.4: Proving Centers of Triangles

9. 1.9.4: Proving Centers of Triangles
Incenter, continued
The incenter is always inside the triangle.
Acute triangle
Obtuse triangle
Right triangle
1.9.4: Proving Centers of Triangles

10. 1.9.4: Proving Centers of Triangles
Incenter, continued
The incenter of a triangle is the center of the circle that connects each of the sides of a triangle. This is known as the circle that inscribes the triangle.
1.9.4: Proving Centers of Triangles

11. 1.9.4: Proving Centers of Triangles
Orthocenter
The altitudes of a triangle are the perpendicular lines from each vertex of the triangle to its opposite side, also called the height of the triangle.
The three altitudes of a triangle are also concurrent.
This point of concurrency is called the orthocenter of the triangle.
1.9.4: Proving Centers of Triangles

12. Orthocenter, continued
The orthocenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle.
The orthocenter is inside acute triangles, outside obtuse triangles, and at the vertex of the right angle of right triangles.
1.9.4: Proving Centers of Triangles

13. Orthocenter, continued
Acute triangle
Obtuse triangle
Right triangle
X is inside the triangle.
X is outside
the triangle.
X is at the vertex of the right angle.
1.9.4: Proving Centers of Triangles

14. 1.9.4: Proving Centers of Triangles
Centroid
The medians of a triangle are segments that join the vertices of the triangle to the midpoint of the opposite sides.
Every triangle has three medians.
The three medians of a triangle are also concurrent.
This point of concurrency is called the centroid of the triangle.
The centroid is always located inside the triangle the distance from each vertex to the midpoint of the opposite side. This is known as the Centroid Theorem.
1.9.4: Proving Centers of Triangles

15. 1.9.4: Proving Centers of Triangles
Theorem
Centroid Theorem
The centroid of a triangle is
the distance from each vertex to the midpoint of the opposite side.
The centroid of this triangle is at point X.
1.9.4: Proving Centers of Triangles

16. 1.9.4: Proving Centers of Triangles
Centroid, continued
The centroid is always located inside the triangle.
The centroid is also called the center of gravity of a triangle because the triangle will always balance at this point.
Acute triangle
Obtuse triangle
Right triangle
1.9.4: Proving Centers of Triangles

17. 1.9.4: Proving Centers of Triangles
Point of Concurrency
Each center serves its own purpose in design, planning, and construction.
Center of triangle
Intersection of…
Circumcenter
Perpendicular bisectors
Incenter
Angle bisectors
Orthocenter
Altitudes
Centroid
Medians
1.9.4: Proving Centers of Triangles

18. Ms. dambreville
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