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Constructing the Orthocenter - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The orthocenter is the point of concurrency of the altitudes in a triangle. A point of concurrency is the intersection of 3 or more lines, rays, segments or planes. The orthocenter is just one point of concurrency in a triangle. The others are the incenter, the circumcenter and the centroid.

One of the four main points of concurrency
of a triangle is the orthocenter.
The orthocenter is where the
three altitudes intersect.

If we look at three different types of triangles,
if I look at an acute triangle
and I drew in one of the altitudes or
if I dropped an altitude as some
might say, if I drew in another altitude,
then this point right here will
be the orthocenter.
I could also draw in the third altitude,
but I know that since this is a point
of concurrency the three altitudes must
intersect there so I only have
to draw two.

If we look at a right triangle, if I were
to draw in an altitude from that vertex,
well, that just happens to be this
leg of this right triangle.
If I drew in the altitude of this triangle,
then I would see -- excuse me, this
side, then this leg would
be its altitude.
And if we drew in this last one from our
90-degree angle, we see that the point
where they are concurrent is right
at the vertex of that right angle.
So in a right triangle your orthocenter
will be at the vertex of the right

And, last, if we look another an obtuse
triangle, we remember in order to find
the altitude of this side we have to extend
that side drop down an altitude
which is outside of our triangle to find
-- and I'm just going to extend
this -- to find the ortho -- to find
the altitude from this vertex, I'm
going to draw a perpendicular
segment through the vertex.
So it looks like it's going to intersect
right over there, and for this third
side I would have to extend it until
we could find our 90-degree angle.

So in an obtuse triangle your orthocenter
will be outside of your triangle.
So expect that on a quiz.

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