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

 Feb 26, 2019

See the following image :



Note that  XY is parallel to ED, YZ is parallel to DC and XZ is parallel to EC .... therefore triangle  XYZ is similar to triangle EDC


And since X,Y  bisect FE and FD in triangle EFD  and XY is parallel to ED then

Triangle FXY is similar to triangle FDE...and....

XY / FY  =  ED / 2FY

XY = ED / 2

2XY = ED

And since XYZ is similar to EDC.....the altitude of EDC = 2*altitude of XYZ



And since XY, ED and IH   are parallel.....the altitude of triangle EDC will be the same altitude as in triangle IHC


So  the altitude of triangle CXY =


altitude of triangle XYZ + altitude of triangle EDC =


altitude of triangle XYZ +  2* altitude of triangle XYZ  =


3*altitude of triangle XYZ


And triangle CXY is on the same base as triangle XYZ....so their areas are to each other as their altitudes


So [ CYX ] = 3 * [ XYZ ] =  3(21)  =  63




cool cool cool

 Feb 26, 2019

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