Mediatrix of a triangle


The bisector of a triangle is that line that, being perpendicular to one of the sides of the triangle, divides the segment or side it cuts into two equal parts.

In other words, the bisector crosses one of the sides of the triangle, forming four right angles or 90º, and dividing said side into two segments of equal length.

The bisector is one of the notable lines in a triangle, along with the bisector.

It should be noted that every triangle has three bisectors, one for each of its sides.

Another important issue to note is that the three bisectors of the triangle intersect at the circumcenter of the figure. This is the midpoint of the circumference that contains the triangle. We can see more clearly what is explained in the figure below where D is the circumcenter.

A relevant characteristic of the circumcenter is also that it is equidistant from the three vertices of the triangle, that is, its distance is the same with respect to each of its vertices.

In the upper image, we observe that the bisectors are those that pass through points E, F, and G, and are points equidistant from the ends of the segments (as we explained previously). Thus, it is true that:

AE = EC, BF = FA, BG = GC

It should be noted that the bisector is a straight line, that is, a sequence of points that extends indefinitely towards a single direction (it does not have curves).

Mediatrix example

Suppose that in the figure below, the line that passes through point D and G is the bisector of segment BC. Likewise, it is known that the DG segment measures 3 meters, the DC segment, 5 meters, and the AB segment, 6 meters. What is the perimeter and area of ​​the triangle?

First, we must remember that we can apply the Pythagorean theorem on the right triangle DGC.

As we see in development, we must remember that BG is equal to GC, so BC is twice GC.

Now if I know the segment AB, you can apply the Pythagorean theorem on the triangle ABC:

So, I can find the perimeter (P) and the area (A) of the triangle, applying Heron's formula and s being the semiperimeter:

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