Assume . Then, since , it follows by the Isosceles Triangle Theorem that their opposite angles are also congruent. Since the measures of the angles of a triangle total , letting :

All three angles have measure , making  equiangular and, as a consequence, equilateral. Therefore, , and the perimeter, or the sum of the lengths of the sides, is

However, the perimeter is given to be 56. Therefore, .

The figure referenced is below:

 is equilateral, so .  is equilateral, so . By the Transitive Property of Congruence, . A quadrilateral with four congruent sides is a parallelogram and a rhombus. However, it is not a rectangle, and, consequently, not a square, since its angles are not right - , an angle of an equilateral triangle, measures . Also, a parallelogram is not a trapezoid. Therefore, the quadrilateral is a parallelogram and a rhombus only.

A midsegment of a triangle - a segment whose endpoints are the midpoints of two sides - is, by the Triangle Midsegment Theorem, parallel to the third side, and is half the length of that side. An equilateral triangle with perimeter 30 has three sides one third this, or 

.

Consequently, the length of each midsegment is half this, or

.

Equilateral triangles have sides of all equal length and angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x, and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = 

Given the height, we can now find the area of the triangle using the equation:

Equilateral triangles have sides of equal length, with angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x, and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = 

Given the height, we can now find the area of the triangle using the equation:

The answer is  .  

To find the area you would first need to find what the length of each side is: 54 divided by 3 is 18 for each side.  

Then you would need to draw in the altitude of the triangle in order to get its height.  Drawing this altitude will create two 30-60-90 degree triangles as shown in the picture.  The longer leg is  times the short leg.  Thus the height is . 

Next we plug in the base and the height into the formula to get 

The formula for the area of an equilateral triangle with side length  is

So, since ,

The formula of the area of an equilateral triangle is  if  is a side.

Since the sides of our triangle have doubled, they have changed from  to . We can substitute  into the equation and solve for the triangle's new area in terms of :

The formula for the area of an equilateral triangle is  if  is the length of one of the triangle's sides.

In this problem, the length of one of the triangle's sides is being tripled, so we can substitute  into the equation for  and solve for the triangle's new area in terms of :

The formula for the area of an equilateral triangle is. For this problem's triangle, , so we can substitute  into the equation for  and solve for the area of the triangle:

At this point, we need to divide by , since the problem asks for half of the triangle's area: