# Special Lines in Triangles: Bisectors, Medians, and Altitudes

TLDRThe script discusses key geometric features of triangles, including perpendicular bisectors, angle bisectors, medians, centroids, altitudes, circumcenters, incenters, and orthocenters. It explains how these lines and points are constructed and their defining properties. For example, the circumcenter is the intersection of perpendicular bisectors and is equidistant from all vertices, while the centroid is where medians intersect and is 2/3 along each median. The summary provides a succinct overview of the triangle geometry covered.

###### Takeaways

- π Describes perpendicular bisectors, angle bisectors, circumcenter, incenter.
- π Explains medians, centroids, altitudes, and orthocenter of a triangle.
- π Perpendicular bisectors intersect at the circumcenter, equidistant from vertices.
- π Angle bisectors intersect at the incenter, equidistant from sides.
- π€ Medians intersect at the centroid, 2/3 from vertex to opposite side.
- π€ Altitudes intersect at the orthocenter, forming right angles.
- π§ Circumcenter is inside acute triangles, outside obtuse triangles.
- π§ Incenter is always inside the triangle.
- π Circumcenter relates to circumcircle, incenter to incircle.
- π Shows how to use theorems to determine angles and segment lengths.

###### Q & A

### What is a perpendicular bisector?

-A perpendicular bisector is a line that is perpendicular to one side of a triangle and bisects (cuts in half) that side.

### What is special about the circumcenter of a triangle?

-The circumcenter is equidistant from the three vertices of the triangle. It is the center of the circumscribed circle, which contains all three vertices.

### Where is the incenter located in a triangle?

-The incenter is always located inside the triangle. It is equidistant from the three sides.

### What is the difference between a median and a bisector?

-A median connects a vertex to the midpoint of the opposite side. A bisector bisects either an angle or a side at a right angle.

### Where is the centroid located?

-The centroid is always located inside the triangle. It is the point where the three medians intersect.

### How can you locate the orthocenter?

-The orthocenter is located at the intersection of the three altitudes of the triangle. The altitudes are perpendicular segments from each vertex to the opposite side.

### What are some special points in a triangle?

-Some special points are the circumcenter, incenter, centroid, and orthocenter. There are also the midpoints of each side.

### How can you prove properties of a triangle?

-You can use theorems about perpendicular bisectors, angle bisectors, medians, and altitudes to prove things like angle measures and segment lengths.

### What is an altitude of a triangle?

-An altitude is a perpendicular segment from a vertex to the opposite side or extension of the opposite side.

### What is a circumscribed circle?

-The circumscribed circle of a triangle passes through all three vertices. Its center is the circumcenter.

###### Outlines

##### π Lines and Points in Triangles

This paragraph introduces different lines that can be drawn in triangles, like perpendicular bisectors, angle bisectors, and medians. It explains how these lines relate to special points in triangles, like the circumcenter, incenter, centroid, and orthocenter. The circumcenter is where perpendicular bisectors intersect, the incenter is where angle bisectors intersect, the centroid is where medians intersect, and the orthocenter is where altitudes intersect.

##### π Checking Comprehension on Triangle Lines

This short paragraph concludes the discussion on lines in triangles. It notes that altitudes form right angles at the vertices, unlike medians. It then prompts the reader to check their understanding of the concepts covered.

###### Mindmap

###### Keywords

##### π‘Triangle

##### π‘Perpendicular bisector

##### π‘Angle bisector

##### π‘Circumcenter

##### π‘Incenter

##### π‘Median

##### π‘Centroid

##### π‘Altitude

##### π‘Orthocenter

##### π‘Right triangle

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###### Transcripts

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