# Equilateral Triangle

## Equilateral Triangle

To construct an equilateral triangle we can use the regular polygon tool and construct a three sided regular polygon. This is a triangle whose angles and sides have equal measures. Also we can construct a line segment, rotate it around the point of the line segment counterclockwise and then again. With this we will have a triangle with sides and angles of equal measures.
Another way to construct an equilateral triangle with circles. First we construct circle A. Then we construct circle B with a radius AB. Then we can construct the intersection of the two circles with point C. When we construct line segments AB, BC, CA we get a triangle. Using the angle measure tool and distance tool to show that all of the measures of the angles and side lengths are equal and therefore it is an equilateral triangle. A more formal proof of this is circle A is congruent to circle B, since they were each formed using the same radius length, AB. Since AB and AC are lengths of radii of circle A, they are equal to each other. Similarly, AB and BC are radii of circle B, and are equal to one another. Therefore the length of by the substitution property. Since congruent segments have equal lengths and triangle ABC is equilateral having three congruent sides. If we make the circles larger or smaller we can observe that the angles of the triangle stay the same and that the lengths increase and decrease but always remain equal to each other.
We can also construct an equilateral triangle from a hexagon. First we can use the regular polygon tool to construct a 6 sided polygon (hexagon). Then we can draw three lines, EF, FG, GE. Using the angle measure and distance length tool we can see that all of the angles measure and that the sides have equal length. If we make the hexagon larger or smaller we can observe that the angles of the triangle stay the same and that the lengths increase and decrease but always remain equal to each other.

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