Triangle Centers Notes worksheet with diagrams and fill-in-the-blank facts for learning about centroid, circumcenter, incenter, and orthocenter.

Problem Analysis:

The task involves filling in the blanks in a chart about the Triangle Centers. The chart lists four triangle centers: Centroid, Circumcenter, Incenter, and Orthocenter, along with their corresponding special segments, pictures, and important facts. The goal is to complete the missing information based on geometric properties.

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Step-by-Step Solution:

#### 1. General Statement at the Top:
The statement at the top says:
> "Triangle Centers - When 3 special segments in a triangle meet at a ________, they create a triangle center or a ________ of __________. The 4 triangle centers are in the chart below."

- Blank 1: The point where three special segments meet is called a point.
- Blank 2: This point is often referred to as a center.
- Blank 3: The term "center" refers to a specific property or characteristic of the triangle.

So, the completed statement is:
> "Triangle Centers - When 3 special segments in a triangle meet at a point, they create a triangle center or a center of symmetry or balance. The 4 triangle centers are in the chart below."

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#### 2. Centroid:
- Special Segment: Medians
- Important Facts:
1. Always inside the triangle: The centroid is always located inside the triangle, regardless of the type of triangle (acute, right, or obtuse).
2. The triangle's center of mass: The centroid divides each median into a ratio of 2:1, with the longer segment being closer to the vertex. It is also the center of mass if the triangle is uniform.

So, the completed facts are:
- Always inside the triangle.
- The triangle's center of mass.

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#### 3. Circumcenter:
- Special Segment: Perpendicular Bisectors
- Important Facts:
1. Inside an acute triangle: For an acute triangle, the circumcenter lies inside the triangle.
2. On a right triangle: For a right triangle, the circumcenter lies at the midpoint of the hypotenuse.
3. Outside an obtuse triangle: For an obtuse triangle, the circumcenter lies outside the triangle.
4. Center of a circumscribed circle: The circumcenter is the center of the circle that passes through all three vertices of the triangle (circumcircle).

So, the completed facts are:
- Inside an acute triangle.
- On a right triangle.
- Outside an obtuse triangle.
- Center of a circumscribed circle.

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#### 4. Incenter:
- Special Segment: Angle Bisectors
- Important Facts:
1. Always inside the triangle: The incenter is always located inside the triangle, regardless of the type of triangle.
2. Center of an inscribed circle: The incenter is the center of the circle that is tangent to all three sides of the triangle (incircle).

So, the completed facts are:
- Always inside the triangle.
- Center of an inscribed circle.

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#### 5. Orthocenter:
- Special Segment: Altitudes
- Important Facts:
1. Inside an acute triangle: For an acute triangle, the orthocenter lies inside the triangle.
2. On a right triangle: For a right triangle, the orthocenter coincides with the vertex of the right angle.
3. Outside an obtuse triangle: For an obtuse triangle, the orthocenter lies outside the triangle.

So, the completed facts are:
- Inside an acute triangle.
- On a right triangle.
- Outside an obtuse triangle.

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Final Completed Chart:

| NAME OF CENTER | SPECIAL SEGMENT | PICTURE | IMPORTANT FACTS |
|----------------------|--------------------------|-------------------------------------------------------------------------|---------------------------------------------------------------------------------|
| Centroid | Medians |  | - Always inside the triangle.<br>- The triangle's center of mass. |
| Circumcenter | Perpendicular Bisectors |  | - Inside an acute triangle.<br>- On a right triangle.<br>- Outside an obtuse triangle.<br>- Center of a circumscribed circle. |
| Incenter | Angle Bisectors |  | - Always inside the triangle.<br>- Center of an inscribed circle. |
| Orthocenter | Altitudes |  | - Inside an acute triangle.<br>- On a right triangle.<br>- Outside an obtuse triangle. |

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Final Answer:

\boxed{
\text{See the completed chart above.}
}