Points D, E, and F are the midpoints of sides BC, CA, and AB, respectively, of triangle ABC. Points X, Y, and Z are the midpoints of EF, FD, and DE, respectively. If the area of triangle XYZ is 21, then what is the area of triangle CXY?

Please.

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To find the area of triangle CXY, we need to understand the relationship between triangle XYZ and triangle CXY.

First, let's recall the concept of midpoints of a triangle. The midpoints of the sides of a triangle divide the triangle into four smaller triangles of equal area.

In this case, triangle ABC is divided into six smaller triangles, including triangle CXY and triangle XYZ. Since points D, E, and F are the midpoints of sides BC, CA, and AB respectively, triangle ABC can be divided into three smaller triangles: triangle CDE, triangle BDF, and triangle AEF.

Now, since points X, Y, and Z are the midpoints of sides EF, FD, and DE respectively, triangle AEF is divided into three smaller triangles: triangle XYZ, triangle XZE, and triangle YXD.

Given that the area of triangle XYZ is 21, we can conclude that each of the smaller triangles formed by the midpoints is also 21 in area, including triangle CDE, triangle BDF, triangle XZE, and triangle YXD.

Now, let's focus on triangle CDE. Triangle CDE is formed by the midpoints of triangle ABC's sides. Since triangle CDE is similar to triangle ABC, we can conclude that the ratio of their areas is the square of the ratio of their corresponding sides.

Let's denote the sides of triangle CDE as c, d, and e, and the sides of triangle ABC as a, b, and c. Since triangle ABC and triangle CDE are similar, we have the following ratios:

c/a = 1/2, d/b = 1/2, and e/c = 1/2.

Taking the squares of these ratios, we get:

(c/a)^2 = (1/2)^2 = 1/4,
(d/b)^2 = (1/2)^2 = 1/4, and
(e/c)^2 = (1/2)^2 = 1/4.

Now, let's consider triangle CXY. Triangle CXY is formed by the midpoints of triangle CDE's sides. Since triangle CXY is similar to triangle CDE, we can conclude that the ratio of their areas is also the square of the ratio of their corresponding sides.

Using the same logic as before, we can see that the sides of triangle CXY are given by:

x/c = 1/2, y/d = 1/2, and c/e = 1/2.

Taking the squares of these ratios, we get:

(x/c)^2 = (1/2)^2 = 1/4,
(y/d)^2 = (1/2)^2 = 1/4, and
(c/e)^2 = (1/2)^2 = 1/4.

Since the ratios of the side lengths of triangle CXY with respect to triangle CDE are the same as the ratios of the side lengths of triangle XYZ with respect to triangle AEF (which we already established to be 1/4), we can conclude that the area of triangle CXY is also 1/4 of the area of triangle XYZ.

Therefore, the area of triangle CXY is 21 * 1/4 = 21/4 = 5.25.

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