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

Lesson 7: Altitudes

# Proof: Triangle altitudes are concurrent (orthocenter)

Showing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter). Created by Sal Khan.

## Want to join the conversation?

• What does concurrent mean, when Sal says the altitudes are concurrent?
• Concurrent lines: Multiple lines that all intersect at a single point on a plane
• If it is a scalene triangle and the altitude of one of the sides forms two congruent angles, what would you say the reason is in you proof? I can't find anything on here but this about altitudes and congruent angles.
• The reason is Alternate interior angles of parrallel lines postulate (AIP postulate)
• So far (in geometry, i'm watching all the geometry vids in order) Sal has talked almost exclusively about triangles. When does he get into squares or even more complex stuff like nonagons?
• Triangles are the base shape in geometry. There are lots of theorems built around triangles. Triangles are the shape with the least sides. Also, every other polygon can be divided into triangles, because it is the base of all polygons. Triangle are very important to learn, especially in geometry, because they will be used in other areas of math too (so are circles too). Nonagons aren't that complex compared to triangles. Triangles are very important to know for a base of real, hard geometry. Nonagons, decagons, and other shapes with bizarre names just have more sides and angles. They really start to get boring after a while. Triangles, on the other hand, are so much more complex and we have lots to learn about them still.
Hope this helped!
• If you continue the altitudes (?) of the middle triangle, it looks like they intersect the larger triangle at some weird point, or is this just the way it is drawn? This seems like a very, very complicated way to find the middle of a triangle.
• The altitudes of the medial triangle end up being the perpendicular bisectors of the larger triangle so they won't necessarily go through any of its vertices. Perpendicular bisectors go through the midpoint of a side and are perpendicular to it but don't have to connect with a vertex.

The video didn't mention it explicitly but it ends up that the circumcenter of a triangle will be the orthocenter of its medial triangle.
• if O is the orthocentre of the trangle ABC and angle BAC is 80 degree then measure of angle BOC is
• If you know BAC, you know ACO and ABO (Draw the triangle to see why). With that, you know what OBC + OCB is (Draw the triangle to see how). And then, Finding BOC is extremely easy.
• So this is the easy way to find the orthocenter? Will this work every time? And how else can you find the orthocenter in simplest terms?
• just make altitudes and the point on which they intersect is orthocenter
• How do I find the altitude of a right triangle?
• An altitude is a perpendicular dropped on a different side from a vertex.
Therefore, the sides making the right angle will be altitudes themselves!!
The third altitude will fall on the hypotenuse.
therefore the orthocentre will be the vertice whose angle is 90 degrees
• 25) From a point in the interior of an equilateral triangle, altitudes to the 3 sides
are drawn. These altitudes have lengths 2, 6, and 4. Find the side length of this
triangle. (Refer to the diagram at the right)
A. B. C. D. E. NOTA
• there was a guy in geometry named Viviani
(1 vote)
• What is the difference between altitudes and medians?
(1 vote)
• "A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.Altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side or an extension of the opposite side."