Can someone help me find the centroid, circumcenter, and the orthocenter of a traingle for the following points:
(-3,6)
(-9,-3)
(4,2)
PLEASE HELP! I worked on this problem for so long and I can't figure it out.
Can someone help me find the centroid, circumcenter, and the orthocenter of a traingle for the following points:
(-3,6)
(-9,-3)
(4,2)
PLEASE HELP! I worked on this problem for so long and I can't figure it out.
Hello, hm_498!
I'll give you a small start . . .
Find the centroid, circumcenter, and the orthocenter of the triangle with vertices:
. . $\displaystyle A(\text{-}3,6),\;B((\text{-}9,\text{-}3),\;C(4,2)$
I must assume you are familiar with slopes and equations of lines . . .
The centroid is the intersection of the medians.
A median joins a vertex to the midpoint of the opposite side.
First, find some of the midpoints of the sides.
Let $\displaystyle D$ be the midpoint of side $\displaystyle BC$. .Then: $\displaystyle D\left(-\frac{5}{2},\,-\frac{1}{2}\right)$
. . Write the equation of $\displaystyle M_1$, the line through $\displaystyle A$ and $\displaystyle D.$
Let $\displaystyle E$ be the midpoint of side $\displaystyle AC$. .Then: $\displaystyle E\left(\frac{1}{2},\,4\right)$
. . Write the equation of $\displaystyle M_2$, the line through $\displaystyle B$ and $\displaystyle E.$
The centroid is the intersection of $\displaystyle M_1$ and $\displaystyle M_2$.
Hints for the rest of the problem:
The circumcenter is the intersection of the perpendicular bisectors of the sides.
The orthocenter is the intersection of the altitudes to the sides.
Here is a diagram of ortho, circum, and centroid.
Press the buttons to make things easier to see.
You can move the red points.
"Mp" stands for "Midpoint"
Red lines are perpendicular.
Magenta Lines are Altitudes.
This is just to give you a visual.
Does it Help?
EDIT: Added Euler's Line (I was bored, ok?)
I would like to mention that they always line (if you have 3 points) on a single line!
It is called "Euler's Line".
He discovered it.
When I was about your age, I proved it using some analytic geometry. The algebra was intense (as you can imagine) but it worked out nicely in the end.
However, I think Euler proved it classically. I wonder how he did that . I have some idea but never been able to. Anybody know?