Math - The Four Lines and Points of Triangles

In this post, I'm going to post some things that I learned about triangles today.

Well, first is that triangles have four basic types of lines, those are:

• Angle Bisector
• Perpendicular Bisector
• Median
• Altitude

And each of these lines creates a point:

• Angle Bisector creates an Incenter
• Perpendicular Bisector creates a Circumcenter
• Median create creates a Centroid
• Altitude creates an Orthocenter

But how can they create those points?

You probably already know that a triangle has three angles and three faces, right?

You can trace those lines on each angle(or face in the case of the Perpendicular Bisector).

So you can get three lines for each type and each of these three lines are going to intersect in a point.

Let's see how they do this:

    1. Angle Bisector / Incenter

First let's understand what bisector means: "a straight line that bisects an angle or a line segment", so we can conclude that an Angle Bisector is a bisector that bisects an angle in two equal angles, let's see an example with a triangle:

Every triangle has three angle-bisector, and when you draw them, it would look like this:

With this, we can now find the Incenter of the triangle, what is the incenter?

The incenter is the central point of a circle inscribed into the triangle, also this point is equidistant(same distance) from all three faces, which is basically the radius of an inscribed triangle.

The incenter would look like this: 

Where the big blue circle is the inscribed circle and the smaller is the incenter.

    2. Perpendicular Bisector / Circumcenter

You already know what a bisector means, but what is a perpendicular bisector?

Let's understand what is a perpendicular line, a perpendicular line is a line that when it intersects another line forms an angle of 90°(right angle), so we can say that a perpendicular bisector always intersects a line forming a right angle in the midpoint.

And just like the angle bisector, the perpendicular bisector also have a point where all the three lines intersect, this point is named circumcenter, and just like the incenter is the center of a circle inscribed into a triangle, the circumcenter is the center of a circumscribed circle(it is a circle around the outside of a figure passing through all of the vertices of the figure, in this case, all the three vertices of the triangle), looking like this: 

    3. Median / Centroid

Well, the median of the triangle can be a little confusing, because it does basically the same as the perpendicular bisector, it divides the triangle segment by its midpoint just like the perpendicular bisector, but in a different way, it starts the division by the angle and not from the segment itself.  

And this intersection forms the centroid, the centroid is very interesting and is also used for physics and astronomy, if you want to know more about it, access this link: https://en.wikipedia.org/wiki/Centroid#Of_a_triangle

    4. Altitude / Orthocenter

The altitude is very easy to understand, as the name already says, the altitude is the line that forms an altitude and a right angle, making it look like this in a triangle:

While the orthocenter is the intersection of all three altitudes.

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And this is the post about the four principal lines and points of a triangle, if you want to know more about them, you can read these links:

Angle Bisector / Incenter

Perpendicular Bisector / Circumcenter

Median / Centroid

Altitude / Orthocenter