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A triangle with three congruent sides is called an equilateral triangle. In an equilateral triangle, not only are all three sides of equal length, but all three interior angles are also equal, each measuring 60 degrees.

Supplemental Knowledge

An equilateral triangle is a special type of triangle where all three sides are of equal length. This unique property leads to several interesting characteristics:

1. Equal Sides:

- In an equilateral triangle, all three sides are congruent, meaning they have the same length.

2. Equal Angles:

- Because the sides are equal, the angles opposite those sides must also be equal. In an equilateral triangle, each interior angle measures exactly 60 degrees.

- This is derived from the fact that the sum of the interior angles in any triangle is always 180 degrees:

\[\text { Sum of angles} = 180^ \circ \]

Since all three angles are equal:

\[\frac { 180^ \circ } { 3} = 60^ \circ \]

3. Symmetry:

- An equilateral triangle has rotational symmetry of order 3 and three lines of reflectional symmetry (each line passes through a vertex and bisects the opposite side).

4. Area and Perimeter Formulas:

- The perimeter \(P\) of an equilateral triangle with side length \(a\) is given by:

\[P = 3a\]

- The area \(A\) can be calculated using the formula:

\[A = \frac { \sqrt { 3} } { 4} a^ 2\]

5. Circumcircle and Incircle:

- An equilateral triangle can be inscribed in a circle (circumcircle), and it also has an incircle that touches all three sides.

- The radius \(R\) of the circumcircle is related to the side length by:

\[R = \frac { a} { \sqrt { 3} } \]

- The radius \(r\) of the incircle is given by:

\[r = \frac { \sqrt { 3} } { 6} a\]

Mastering geometric concepts such as those related to equilateral triangles can greatly expand your mathematical knowledge and ability. UpStudy provides additional assistance with geometry as well as other topics, should further help be needed.