Relationships between the Centers – Irrationality
The orthocenter, the centroid and the circumcenter of a non-equilateral triangle are aligned; that is to say, they belong to the same straight line, called line of. A median of a triangle is the segment from a vertex to the midpoint of the opposite side. Obtuse Triangle, The centroid of a obtuse triangle is inside of the triangle. The ORTHOCENTER(H) of a triangle is the common intersection of the three lines What relationships can you find among G, H, C, and I or subsets of them?. If G is the centroid of the triangle, that relation follows from. A O → + B O → + C O → = A G → + B G → + C G → + 3 G O → = 3 G O → = H O →.
And then we can use the exact same argument for this one over here. That by itself is interesting.Centroid, Circumcenter, Incenter, Orthocenter
But we know that if two triangles are congruent, all of their corresponding sides and angles are going to be congruent. So for example, if we know the measure of this angle is blue, the corresponding angle on this triangle is also going to have the same measure.
I'll just make it with that same blue angle. And we know if this angle right over here is magenta, the corresponding angle on triangle AFG is also going to have that same measure.
And I'll just mark it with that magenta again. Now we also know from our properties of vertical angles, that whatever angle measure AFG is, DGC is going to have the same measure, because they are vertical angles.
Orthocenter, Centroid, Circumcenter and Incenter of a Triangle
But we know whatever angle measure this is, this triangle, triangle CDG, is congruent to triangle EDG, so corresponding angles have to be congruent. So this angle is this magenta measure, then this angle also has to be the magenta measure.
And then once again, you see vertical angles. If this is magenta, then this is also going to have that same measure. And if this has a measure, then that is also going to have the same measure. So by using a little bit of argument of congruent triangles, corresponding angles are going to be congruent, and vertical angles, we can see that all of these inner angles right over here are going to have to be the same measure.
And I'm using that with this little magenta arc right over there. Now all of these triangles, that we split this triangle into, they all have a 90 degree angle, they all have a magenta arc. So whatever's left over is going to be minus 90 minus magenta. Or it's really 90 minus this magenta angle right over here. And that's what this blue angle must be.
Centers of Triangle_IMMICH
This blue angle is essentially 90 minus the magenta angle. The blue angle is 90 minus the magenta angle. And so that angle must be the third angle for all of these. So once again, this blue angle, there's going to be 90 minus the magenta angle, or minus the magenta minus the So that's going to be a blue angle.
Essentially what we're saying is, if you know two angles of a triangle, it forces what the other angle is going to be. We already know that all six of these triangles have two angles in common-- the 90 degree angle and the magenta angle.
So the third angle must be the same as well. Thus, the radius of the circle is the distance between the circumcenter and any of the triangle's three vertices. It is found by finding the midpoint of each leg of the triangle and constructing a line perpendicular to that leg at its midpoint.
Where all three lines intersect is the circumcenter. The circumcenter is not always inside the triangle.
In fact, it can be outside the triangle, as in the case of an obtuse triangle, or it can fall at the midpoint of the hypotenuse of a right triangle. See the pictures below for examples of this. You see that even though the circumcenter is outside the triangle in the case of the obtuse triangle, it is still equidistant from all three vertices of the triangle.
If you have Geometer's Sketchpad and would like to see the GSP construction of the circumcenter, click here to download it. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side. An altitude of the triangle is sometimes called the height. Remember, the altitudes of a triangle do not go through the midpoints of the legs unless you have a special triangle, like an equilateral triangle.
Like the circumcenter, the orthocenter does not have to be inside the triangle. Check out the cases of the obtuse and right triangles below.
Common orthocenter and centroid
In the obtuse triangle, the orthocenter falls outside the triangle. In a right triangle, the orthocenter falls on a vertex of the triangle.
- Relationships between the Centers
- Orthocenter, Centroid, Circumcenter and Incenter of a Triangle
If you have Geometer's Sketchpad and would like to see the GSP construction of the orthocenter, click here to download it. It is the point forming the origin of a circle inscribed inside the triangle.
Like the centroid, the incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle. The radius of the circle is obtained by dropping a perpendicular from the incenter to any of the triangle legs. It is pictured below as the red dashed line.