The figure referenced is below:

For the sake of simplicity, assume that the square has sides of length 4. The following reasoning is independent of the actual lengths, and the reason for choosing 4 will become apparent in the explanation.

 and  are midpoints of their respective sides, so , making  the hypotenuse of a triangle with legs of length 2 and 2. Therefore,

.

Also, , and since  is the midpoint of , . , making  the hypotenuse of a triangle with legs of length 1 and 4. Therefore, 

, so 

The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular, 

.

Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,

The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular, 

.

Their corresponding sides are in proportion, so, setting the ratios of the long legs to the short legs equal to each other,

By the Pythagorean Theorem. 

The proportion statement becomes

The measure of the angle formed by the two shorter sides of a triangle can be determined to be acute, right, or obtuse by comparing the sum of the squares of those lengths to the square of the length of the opposite side. We compare:

; it follows that  is obtuse, and has measure greater than 

The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular, 

.

Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,

Each right triangle is a  triangle, making each triangle isosceles by the Converse of the Isosceles Triangle Theorem.

Since  and  are the right triangles, the legs are , and the hypotenuses are .

By the  Theorem,  and .

, so  and subsequently, .

The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller, similar triangles. 

The similarity ratio of  to  can be found by determining the ratio of one pair of corresponding sides; we will use the short leg of each,  and .

 is also the long leg of , so its length can be found using the Pythagorean Theorem:

The similarity ratio is therefore

.

The ratio of the areas is the square of this ratio:

 - that is, 144 to 25.

By the Pythagorean Theorem,

The sum of the measures of the angles of a triangle is , so:

This is a  triangle, so its legs  and  are congruent. The quantities are equal.

The area of a triangle is half the product of its height and its base; in a right triangle, the legs, being perpendicular, can serve as these quantites.

The triangle in the diagram has area 

 square inches.

An isosceles right triangle has two legs of the same length, which we will call . The area of that triangle, which is the same as that of the one in the diagram, is therefore 

 inches.