Triangle Center Stuff for Cynaya!
Circumcenter (perpendicular bisectors)
The circumcenter is the intersection of the perpendicular bisectors of each of the sides of the triangle
*Note*
-The circumcenter is inside the triangle if the triangle is acute (all triangle angles are acute)
-The circumcenter is outside the triangle if the triangle is obtuse (the triangle has an obtuse angle)
-The circumcenter is on the hypotenuse of the triangle if the triangle is right (the triangle has a right angle)
*Try testing these by moving the points of the triangle around!
A perpendicular bisector is a line that bisects(cuts in half) the line it is perpendicular to.
Orthocenter (Altitudes)
The orthocenter is the intersection of the altitudes of the triangle.
*Note*
The orthocenter is only inside the triangle if the triangle is acute (all angles are less than 90)
The altitude of a triangle is also known as the height of a triangle. The altitude is a line that goes through a point of the triangle and is perpendicular to the side opposite of the point it goes through.
This one is the most confusing, I know... I shaded in the actual triangle extra dark so it's easier to distinguish from all the crazy lines in this one.
Incenter (angle bisectors)
The incenter of a triangle is the interact point of all the angle bisectors of the angles in the triangle.
*Note*
The incenter will ALWAYS be inside the triangle
An angle bisector is a line that bisects (cuts in half) the angle it's going through.
Notice! The angle bisectors don't always go through the midpoints of the sides of the triangle! It can happen but isn't always true.
Centroid (Medians)
The centroid is the intersection of each of the medians of the triangle
*Note the centroid is ALWAYS inside the triangle
The median of a triangle is the line that goes through the midpoint of a side of your triangle and the point opposite that side. (i.e. The median of AB goes through point C since opposite AB)
Notice! The median doesn't bisect the angles of the triangle. Look at C and the median of AB
*A note about medians and centroids
For the medians of triangles, the centroid basically splits the median into a 2/3 chunk and a 1/3 chunk.
Let me explain...
The median is the segment from the midpoint to the opposite point. The centroid cuts that median into 2 segments. The short section will end up being half of the long section of the median.
In other words, 2 small chuncks=1 large chunk.
So the length of the smaller chunk is 1/3 of the length of the whole median. And the larger chunk is 2/3 of the whole median.
We can go over this more next time if this is still confusing.