{"status": "success", "data": {"description_md": "Consider a tangential quadrilateral $ABCD$ with $AB = 4, BC=9, CD=13,$ and $DA = 8$. Let $I$ be the center of the inscribed quadrilateral. Draw perpendiculars from $I$ to $AB, BC, CD, DA$, and let them be $H_A,H_B,H_C,H_D$ respectively. If the length of $AH_A$ to maximize $[ABCD]^2$ can be written as $\\frac{m}{n}$, where $\\gcd(m,n) = 1$, find $m + n$.", "description_html": "<p>Consider a tangential quadrilateral <span class=\"katex--inline\">ABCD</span> with <span class=\"katex--inline\">AB = 4, BC=9, CD=13,</span> and <span class=\"katex--inline\">DA = 8</span>. Let <span class=\"katex--inline\">I</span> be the center of the inscribed quadrilateral. Draw perpendiculars from <span class=\"katex--inline\">I</span> to <span class=\"katex--inline\">AB, BC, CD, DA</span>, and let them be <span class=\"katex--inline\">H_A,H_B,H_C,H_D</span> respectively. If the length of <span class=\"katex--inline\">AH_A</span> to maximize <span class=\"katex--inline\">[ABCD]^2</span> can be written as <span class=\"katex--inline\">\\frac{m}{n}</span>, where <span class=\"katex--inline\">\\gcd(m,n) = 1</span>, find <span class=\"katex--inline\">m + n</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 7, "problem_name": "TxO Math Bowl 2024 - Guts Contest - Set 8 Problem 1", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}