Educational worksheet on triangle centers with diagrams and handwritten notes.

The image shows a worksheet titled "Centers of Triangles," which explains the properties and characteristics of four key triangle centers: Circumcenter, Incenter, Centroid, and Orthocenter. Below, I will explain each center in detail and solve the problem by filling in the missing information.

---

1. Circumcenter

- Created by: Perpendicular Bisectors
- Important Facts:
- The Circumcenter is equidistant from each vertex of the triangle.
- It is the center of the circle that passes through all three vertices of the triangle (the circumcircle).
- In the diagram:
- Perpendicular bisectors are drawn for each side of the triangle.
- The point where these bisectors intersect is labeled as the Circumcenter.
- Key property: \( AM = BM = CM \), where \( M \) is the Circumcenter.

---

2. Incenter

- Created by: Angle Bisectors
- Important Facts:
- The Incenter is equidistant from each side of the triangle.
- It is the center of the circle that is tangent to all three sides of the triangle (the incircle).
- In the diagram:
- Angle bisectors are drawn for each angle of the triangle.
- The point where these bisectors intersect is labeled as the Incenter.
- Key property: The Incenter is always inside the triangle.

---

3. Centroid

- Created by: Medians
- Important Facts:
- A median is created by a vertex connected to the midpoint of the opposite side.
- The Centroid is the point where all three medians of the triangle intersect.
- In the diagram:
- Medians are drawn from each vertex to the midpoint of the opposite side.
- The point where these medians intersect is labeled as the Centroid.
- Key property: The Centroid divides each median into a ratio of \( 2:1 \), with the longer segment being closer to the vertex.

---

4. Orthocenter

- Created by: Altitudes
- Important Facts:
- An altitude is created by a vertex connected to the opposite side so that it is perpendicular to that side.
- The Orthocenter is the point where all three altitudes of the triangle intersect.
- In the diagram:
- Altitudes are drawn from each vertex, perpendicular to the opposite side.
- The point where these altitudes intersect is labeled as the Orthocenter.
- Key property: The location of the Orthocenter varies depending on the type of triangle:
- For an acute triangle, the Orthocenter is inside the triangle.
- For a right triangle, the Orthocenter is at the right angle vertex.
- For an obtuse triangle, the Orthocenter is outside the triangle.

---

Filling in the Missing Information

Based on the above explanations, we can fill in the missing parts of the worksheet:

#### Orthocenter Section:
- Created by: Altitudes
- Important Facts:
- An altitude is created by a vertex connected to the opposite side so that it is perpendicular to that side.

---

Final Answer:

\[
\boxed{\text{Altitudes}}
\]

This completes the explanation and solution for the worksheet. Each section now has its corresponding details filled in accurately.