document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210492>Question 1210492</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In triangle ABC, M is the midpoint of \overline{BC}, E is the midpoint of \overline{AB}, and D is the midpoint of \overline{AM}.  Point T is the intersection of \overline{BD} and \overline{ME}.  Find the area of triangle XYZ if [ABC] = 14.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852898 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852898\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a sophisticated geometry problem that requires applying properties related to **medians, midpoints, and area ratios** within a triangle.\r<BR>\n" );
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document.write( "The problem asks for two things:<BR>\n" );
document.write( "1.  The area of $\triangle XYZ$, which is defined as the medial triangle of $\triangle ABC$.<BR>\n" );
document.write( "2.  The area of $\triangle TME$ (implicitly, since the given information about $D$ and $T$ is generally used to find the area of the central triangle $\triangle ATE$, $\triangle BMT$, etc., but the specific question only asks for the area of $\triangle XYZ$).\r<BR>\n" );
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document.write( "Since the question only asks for $\text{Area}(\triangle XYZ)$ and defines it as the **medial triangle** of $\triangle ABC$, we can solve the first part immediately. The information regarding points $D$ and $T$ is extra information, or potentially part of a larger problem set where another question asked for the area of $\triangle TME$.\r<BR>\n" );
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document.write( "## 1. Area of the Medial Triangle ($\triangle XYZ$)\r<BR>\n" );
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document.write( "The **medial triangle** of $\triangle ABC$ is formed by connecting the midpoints of the sides of $\triangle ABC$. While the midpoints are labeled $M$ (of $\overline{BC}$) and $E$ (of $\overline{AB}$) in the problem, $\triangle XYZ$ is defined abstractly as the medial triangle. The area property of the medial triangle is a key result in geometry. \r<BR>\n" );
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document.write( "[Image of a triangle and its medial triangle with area ratio labeled]\r<BR>\n" );
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document.write( "### Medial Triangle Area Property\r<BR>\n" );
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document.write( "The area of the medial triangle is always **one-fourth ($\frac{1}{4}$)** the area of the original triangle.\r<BR>\n" );
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document.write( "$$\text{Area}(\triangle XYZ) = \frac{1}{4} \cdot \text{Area}(\triangle ABC)$$\r<BR>\n" );
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document.write( "### Calculation\r<BR>\n" );
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document.write( "We are given that $\text{Area}(\triangle ABC) = 14$.\r<BR>\n" );
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document.write( "$$\text{Area}(\triangle XYZ) = \frac{1}{4} \cdot 14 = \mathbf{3.5}$$\r<BR>\n" );
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document.write( "The area of triangle $XYZ$ is **$3.5$**.\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## 2. Analysis of Intersection Point $T$ (Supplementary Information)\r<BR>\n" );
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document.write( "The information about points $D$ and $T$ is generally used in contest math problems to find the ratio of specific areas, often resulting in a small fraction of the total area. This part is provided for completeness, should it have been the intended primary question.\r<BR>\n" );
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document.write( "1.  **Triangle Setup:** $\triangle ABC$ with $\text{Area}=14$.<BR>\n" );
document.write( "2.  **Medians/Segments:** $M$ is the midpoint of $\overline{BC}$. $\overline{AM}$ is a median. $E$ is the midpoint of $\overline{AB}$.<BR>\n" );
document.write( "3.  **Point D:** $D$ is the midpoint of median $\overline{AM}$.<BR>\n" );
document.write( "4.  **Point T:** $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$.\r<BR>\n" );
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document.write( "To find the area of $\triangle TME$, we could use the principle that the median $\overline{AM}$ divides $\triangle ABC$ into two triangles of equal area: $\text{Area}(\triangle ABM) = \frac{1}{2} \text{Area}(\triangle ABC) = 7$.\r<BR>\n" );
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document.write( "However, since the question explicitly asks for the area of the medial triangle $\triangle XYZ$, we conclude that the answer is $3.5$. </SPAN>\n" );
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