{"status": "success", "data": {"description_md": "Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\\triangle{ABP}$, $\\triangle{BCP}$, $\\triangle{CDP}$, and $\\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral? <br><center><img class=\"problem-image\" alt='[asy] unitsize(120); pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3); draw(A--B--C--D--cycle); dot(P); defaultpen(fontsize(10pt)); draw(A--P--B); draw(C--P--D); label(\"$A$\", A, W); label(\"$B$\", B, W); label(\"$C$\", C, E); label(\"$D$\", D, E); label(\"$P$\", P, N*1.5+E*0.5); dot(A); dot(B); dot(C); dot(D); [/asy]' class=\"latexcenter\" height=\"218\" src=\"https://latex.artofproblemsolving.com/4/a/7/4a7552d7af26cb2be88658376dd5934a70ef16c6.png\" width=\"245\"/></center>\n\n$\\textbf{(A) }100\\sqrt{2}\\qquad\\textbf{(B) }100\\sqrt{3}\\qquad\\textbf{(C) }200\\qquad\\textbf{(D) }200\\sqrt{2}\\qquad\\textbf{(E) }200\\sqrt{3}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Square  <span class=\"katex--inline\">ABCD</span>  has side length  <span class=\"katex--inline\">30</span> . Point  <span class=\"katex--inline\">P</span>  lies inside the square so that  <span class=\"katex--inline\">AP = 12</span>  and  <span class=\"katex--inline\">BP = 26</span> . The centroids of  <span class=\"katex--inline\">\\triangle{ABP}</span> ,  <span class=\"katex--inline\">\\triangle{BCP}</span> ,  <span class=\"katex--inline\">\\triangle{CDP}</span> , and  <span class=\"katex--inline\">\\triangle{DAP}</span>  are the vertices of a convex quadrilateral. What is the area of that quadrilateral? <br/><center><img class=\"latexcenter\" alt=\"[asy] unitsize(120); pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3); draw(A--B--C--D--cycle); dot(P); defaultpen(fontsize(10pt)); draw(A--P--B); draw(C--P--D); label(&#34;$A$&#34;, A, W); label(&#34;$B$&#34;, B, W); label(&#34;$C$&#34;, C, E); label(&#34;$D$&#34;, D, E); label(&#34;$P$&#34;, P, N*1.5+E*0.5); dot(A); dot(B); dot(C); dot(D); [/asy]\" height=\"218\" src=\"https://latex.artofproblemsolving.com/4/a/7/4a7552d7af26cb2be88658376dd5934a70ef16c6.png\" width=\"245\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }100\\sqrt{2}\\qquad\\textbf{(B) }100\\sqrt{3}\\qquad\\textbf{(C) }200\\qquad\\textbf{(D) }200\\sqrt{2}\\qquad\\textbf{(E) }200\\sqrt{3}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2018 AMC 12B Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/18_amc12B_p14", "prev": "/problem/18_amc12B_p12"}}