{"status": "success", "data": {"description_md": "Let $\\triangle ABC$ have side lengths $10$, $17$, and $21$. Suppose that square $ABX_1X_2$ is constructed such that $X_1$ and $C$ lie on the opposite side of $\\overline{AB}$. Construct $BCY_1Y_2$, and $CAZ_1Z_2$ similarly.\n\nDenote the pairwise intersections of $\\overline{X_1X_2}$, $\\overline{Y_1Y_2}$, $\\overline{Z_1Z_2}$ as $J$,$K$,$L$, and denote pairwise intersections of $\\overline{X_1Y_2}$, $\\overline{Y_1Z_2}$, $\\overline{Z_1X_2}$ as $P$,$Q$,$R$.\n\nFind the shared area between triangles $JKL$ and $PQR$.\n\nNote: $\\overline{K_1K_2}$ denotes the infinite extension of line segment $K_1K_2$ on both sides.", "description_html": "<p>Let <span class=\"katex--inline\">\\triangle ABC</span> have side lengths <span class=\"katex--inline\">10</span>, <span class=\"katex--inline\">17</span>, and <span class=\"katex--inline\">21</span>. Suppose that square <span class=\"katex--inline\">ABX_1X_2</span> is constructed such that <span class=\"katex--inline\">X_1</span> and <span class=\"katex--inline\">C</span> lie on the opposite side of <span class=\"katex--inline\">\\overline{AB}</span>. Construct <span class=\"katex--inline\">BCY_1Y_2</span>, and <span class=\"katex--inline\">CAZ_1Z_2</span> similarly.</p>&#10;<p>Denote the pairwise intersections of <span class=\"katex--inline\">\\overline{X_1X_2}</span>, <span class=\"katex--inline\">\\overline{Y_1Y_2}</span>, <span class=\"katex--inline\">\\overline{Z_1Z_2}</span> as <span class=\"katex--inline\">J</span>,<span class=\"katex--inline\">K</span>,<span class=\"katex--inline\">L</span>, and denote pairwise intersections of <span class=\"katex--inline\">\\overline{X_1Y_2}</span>, <span class=\"katex--inline\">\\overline{Y_1Z_2}</span>, <span class=\"katex--inline\">\\overline{Z_1X_2}</span> as <span class=\"katex--inline\">P</span>,<span class=\"katex--inline\">Q</span>,<span class=\"katex--inline\">R</span>.</p>&#10;<p>Find the shared area between triangles <span class=\"katex--inline\">JKL</span> and <span class=\"katex--inline\">PQR</span>.</p>&#10;<p>Note: <span class=\"katex--inline\">\\overline{K_1K_2}</span> denotes the infinite extension of line segment <span class=\"katex--inline\">K_1K_2</span> on both sides.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "TxO Math Bowl 2024 - Team Contest - Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024team-p11", "prev": "/problem/txo2024team-p09"}}