{"status": "success", "data": {"description_md": "In $\\triangle ABC$, let points $D$ and $E$ be on $AC$ such that $BD$ and $BE$ trisect $\\angle ABC$ and $D$ is between $A$ and $E$. Let $F$ be a point on the extension of $CA$ such that $\\angle FBA = \\frac{1}{3}(180^{\\circ} - \\angle ABC)$. Given that $BC = 6, EC = 3, DE = 2, FD = 4\\sqrt{3}$, compute the length of $BF$.", "description_html": "<p>In <span class=\"katex--inline\">\\triangle ABC</span>, let points <span class=\"katex--inline\">D</span> and <span class=\"katex--inline\">E</span> be on <span class=\"katex--inline\">AC</span> such that <span class=\"katex--inline\">BD</span> and <span class=\"katex--inline\">BE</span> trisect <span class=\"katex--inline\">\\angle ABC</span> and <span class=\"katex--inline\">D</span> is between <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">E</span>. Let <span class=\"katex--inline\">F</span> be a point on the extension of <span class=\"katex--inline\">CA</span> such that <span class=\"katex--inline\">\\angle FBA = \\frac{1}{3}(180^{\\circ} - \\angle ABC)</span>. Given that <span class=\"katex--inline\">BC = 6, EC = 3, DE = 2, FD = 4\\sqrt{3}</span>, compute the length of <span class=\"katex--inline\">BF</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "TxO Math Bowl 2024 - Individuals B - Problem 5", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024indivsB-p06", "prev": "/problem/txo2024indivsB-p04"}}