This is demonstrated by the inscribed angle on a circle.  An inscribed angle is an angle whose vertex is on the circumference of the circle and whose sides extend as chords of the circle.  This theorem is proven to be true and can be used to solve for angles of inscribed figures in circles.  This theorem is portrayed in the figure below.

Since  is the radius, we can extend it to represent the diameter which is .

We know that a square is made up of two right triangles.  So the diagonal must be a product of the Pythagorean Theorem; .  We should let and since we are working with a square we know .  We can simplify this formula to be

If we take the square root of each side we get

The area of a square is just .  So the area of this square in terms of  is

Line  is perpendicular to line  and is 90 degrees.  We know this because line  intercepts an arc of 180 degrees.  An inscribed angle is half the measure of the intercepted arc. Therefore this must be a right triangle.

Since we are finding the side lengths of a square, we really only need to find one side since all sides are congruent in a square.  Recall that the formula for the area of a triangle is .  We can find the area of the square by finding the area of the triangle and subtracting the area of the square.  Having the area of the square will just be one of the side lengths squared, so if we take the square root of the area of the square, we will have our answer.

First, consider the small triangle in the upper corner.  This is shown below:

The area of this triangle would be 

We will consider the lower two triangles combined to be the second triangle:

The area of this triangle would be

And then we need to consider a third area which is the square

The area of the square would be 

To get the total area of the triangle, we can sum all three areas.

We can sub in our areas and solve for to solve this problem

 (multiply by 2 to get rid of fractions)

To be congruent, triangles must have all corresponding angles and sides be of the same measure.  Corresponding sides of triangles are sides of the same measure in the same positions on different triangles.  This is shown in red on the two triangles below.  Corresponding angles of triangles are angles of the same measure in the same position on different triangles.  This is shown in blue on the two triangles below.

Shapes, in general, are similar when they are the same shape but have different sizes.  If they were the exact same size, they would be congruent.  To have the same shape, even when the sizes differ, the shape needs to have the same angles.  Take the two triangles below for example, they have equal angles but are different sizes.  Therefore these triangles are similar.

If two right triangles have a congruent leg and hypotenuse we can say they have two congruent sides.  Since both of these triangles are right triangles, we also know that they have a congruent angle (their 90-degree angle).  So these two triangles have two pairs of corresponding congruent sides and one pair of corresponding congruent angles.  By SAS Theorem, these right triangles are congruent.  When using this fact that two right triangles are congruent when they have a congruent hypotenuse and a congruent leg, this is called the HL Theorem.

The Side-Side-Side (SSS) Theorem states that two triangles are congruent if the three corresponding sides of each triangle are congruent.  If you think about it, in order for all three corresponding sides to be equal they must fit together at the same angles as well making these triangles congruent.  A detailed proof based on Philoh’s approach is below for a better explanation.  Note that there are stronger proofs than this for this theorem, but this proof is best visually for this level of understanding of the theorem.

Proof: 

(We want to show that if all three sides of two triangles are equal, then these two triangles are congruent)

Consider two triangles  and .  Assume the following:

Since  we are able to rotate triangle  to coincide  and .  We will just call this common line segment  for simplification.

Now we will draw a line from vertex  to vertex .  This gives us two isosceles triangles.  Recall from the definition of an isosceles triangle that the two adjacent sides and angles are equivalent.  So  and .

We can then deduce that . We now know that at least two sides of the two given triangles are equal and the angle enclosed by these sides is congruent between the two triangles. By SAS Theorem, these two triangles are congruent.

Explanation: For the explanation follow the detailed proof below:

In order to solve for  and  below, we need to use the fact that similar triangles are proportional.  This allows us to set up ratios of the lengths of the sides to solve for the unknown variables:

To solve for :

So we can set up the following ratios

  by cross-multiplying to get rid of the fractions

So 

To solve for :

So we can set up the following ratios

 by cross-multiplying to get rid of the fractions

So