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

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Constructing the Orthocenter - Problem 1

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

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The orthocenter of a triangle is the point of concurrency of the three altitudes of that triangle. Recall that altitudes are lines drawn from a vertex, perpendicular to the opposite side. So, find the altitudes. It is sufficient to find only two of the altitudes because the point of intersection of two altitudes of a triangle is the same as the intersection with the third. This point of intersection is the point of concurrency, and the orthocenter of the triangle.

A common construction problem once you know how to construct an altitude is to construct the orthocenter of a triangle. Notice what type of triangle I’ve drawn here it’s an obtuse triangle which means our orthocenter is going to be outside of our circle. But back up what is the orthocenter?

Well the orthocenter is the point of concurrency of the three altitudes, so if I find two of those altitudes then I’ll have my orthocenter. So let’s grab our compass and I’m going to construct an altitude from this point to this opposite side and if I look at it closely I see that I’m not going to have enough room to swing an arc from that vertex. So I’m going to extend this side using my straightedge.

So I’m going to make sure that this side is well extended so that way when I swing my arc I’m going to have two clear points of intersection. So it looks like I need to make my compass a little larger, see if this will work. Okay a little bit larger than that and I’m going to swing an arc so I have two points of intersection. Okay so I’ve got one two points of intersection. I already have one point up here so I just need another point down below here.

Let’s make this a little smaller make sure that it’s at least half of this line segment between those two points of intersection. I’m going to swing an arc here from the other endpoint here, I'm going to swing another arc and I have enough information to draw in my altitude. So I’m going to connect the vertex with that point of intersection.

So we can mark this as a right angle. So we only need one more and I’m going to chose and altitude from this point to that opposite side so we go back to our compass. We are going to swing another arc and I’m going to extend this just a little bit, so I’m going to swing an arc and we have our two points of intersection we are just doing the same exact process here.

I’m going to swing two arcs, one from each of these endpoints so there’s my first arc and here’s my second arc. So now I can connect this vertex with that point of intersection and it should intersect this altitude at some point.

So if I lined these two up and if I connect them I have constructed my orthocenter which is right here and again to you that if I construct the altitude from this point that it would pass through this point. The way that you would have to do that is you would have to extend this line let me put a dotted line here because I’m not actually going to do that. We did that to extend this line and when you will swing an arc to more and I gain to you that it would look something kind of like this. I get everything lined up. So that will be the third altitude.

All these are concurrent which means they intersect at one point.

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