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### Course: Geometry (all content)>Unit 4

Lesson 8: Bringing it all together

# Review of triangle properties

Comparing perpendicular bisectors to angle bisectors to medians to altitudes. Created by Sal Khan.

## Want to join the conversation?

• What are the properties of the orthocentre?
• The orthocenter is the point where all three altitudes of a triangle meet. It doesn't have any other special properties on its own, but if you check out the Euler Line video, you can find more neat things about it.
• What is the difference between a circumcenter and a orthocenter?
• Circumcenter is the intersection of the perpendicular bisectors of a triangle, while orthocenter means the intersection of the altitudes.
• What is the difference between orthocenter, cenroid, incenter and circumcenter? I'm confused between all of them especially orthocenter and centeroid??
(1 vote)
• Those are all what are called point of concurrency, meaning the intersection of multiple lines. Given a triangle,
the orthocenter is the point where the altitudes meet (lines drawn from each vertex that are perpendicular to each side);
the centroid is where the medians meet (lines drawn from each vertex to the midpoint of the opposite side);
the circumcenter is where the perpendicular bisectors meet (lines drawn from the midpoint of each side that are perpendicular to that side);
and the incenter is where the angle bisectors meet (lines from each vertex that divide the angles in half.
The centroid is also the center of mass or balancing pont of the triangle.
The incenter is the center point of a circle that can be inscribed in the triangle (just touches each side and is contained within the triangle)
The circumcenter is the center of a circle that circumscribes the triangle (is drawn just outside the triangle and just touches the three vertices of the triangle.
• How can you find the orthocenter from given vertices?
• First you need the perpendicular slope of the line segment that the altitude goes to. Then, you can do cross products. So if the perpendicular slope is -5/3 and the vertice at (-1,2) you would do -5/3=y-2 over x+1. That will get you the equation 3y-6=-5x-5. You then put it into standard form which would be 5x+3y=1.
• 'ortho'(In orthocenter) means what?
• "Ortho" is a Greek root that means "straight", "right", or "correct"
• Is the median ever the perpendicular bisector of an angle
(1 vote)
• Careful with using terms. You don't have perpendicular bisectors of angles. When we bisect an angle, the line dividing the angle in half is called the "angular bisector".

"Perpendicular bisectors" bisect sides (that is to say, line segments) at a point that we can label as the midpoint of that side. In a triangle, 3 sides means 3 different perpendicular bisectors. Since each perpendicular bisector is perpendicular to the side it divides in half, it does NOT necessarily have to pass through the vertex opposite the side. "Medians" by definition DO pass through the vertex opposite the side.

When perpendicular bisector DOES pass through the vertex, yes, the median line passing through THAT vertex, and the perpendicular bisector of THAT side opposite the vertex, are one and the same. (And guess what. They coincide with the angular bisector of the angle at that vertex, too!) This can happen in isocèles triangles, as well as equilateral triangles. But careful again, in the case of the isocèles triangle, the other two medians of the triangle will NOT coincide with the other two perpendicular bisectors of the triangle. it may seem quite nitpicky, but as Sal often emphasizes in his videos, you have to try to be precise about what you're talking about. Hope this helped.
• Don't perpendicular lines have to bisect the opposite angle?