<p>Let <img alt="$\triangle ABC$" height="18" src="https://latex.artofproblemsolving.com/8/c/3/8c3a2d2224f7d163b46d702132425d47828bf538.png" width="61" /> have side lengths <img alt="$AB=5$" height="13" src="https://latex.artofproblemsolving.com/7/f/f/7ff29028b2432b4e2cf3d924ac82114813de066d.png" width="61" />, <img alt="$BC=9$" height="12" src="https://latex.artofproblemsolving.com/2/f/8/2f8bf6a33fda7064bc18de335471565522d1147a.png" width="61" />, <img alt="$CA=10$" height="14" src="https://latex.artofproblemsolving.com/7/f/d/7fdba30e1f5eae855a7b2f6b36836d8a447ac63f.png" width="72" />. The tangents to circumcircle of <img alt="$\triangle ABC$" height="18" src="https://latex.artofproblemsolving.com/8/c/3/8c3a2d2224f7d163b46d702132425d47828bf538.png" width="61" /> at <img alt="$B$" height="12" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="14" /> and <img alt="$C$" height="12" src="https://latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png" width="14" /> intersect at point <img alt="$D$" height="14" src="https://latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png" width="17" />, and <img alt="$\overline{AD}$" height="15" src="https://latex.artofproblemsolving.com/c/f/b/cfbb4c52e011ceed6e813118a7d59e931af444a2.png" width="29" /> intersects the circumcircle at <img alt="$P \neq A$" height="18" src="https://latex.artofproblemsolving.com/c/7/2/c725c31d3b02df772959c7a80491d1ae896abf1f.png" width="54" />. The length of <img alt="$AP$" height="13" src="https://latex.artofproblemsolving.com/e/3/b/e3bb4eeb347e74e025fd175c22f1f7275af0555e.png" width="27" /> is equal to <img alt="$\frac{m}{n}$" height="33" src="https://latex.artofproblemsolving.com/2/7/f/27f583697295fc6f723862c13228ee55107c33ca.png" width="18" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime integers. Find <img alt="$m + n$" height="12" src="https://latex.artofproblemsolving.com/5/3/b/53bf1afcb274a3bb84338109fffec8a1f1c02008.png" width="48" />.</p> <h2>Diagram</h2> <p><img alt="[asy] import olympiad; unitsize(15); pair A, B, C, D, E, F, P, O; C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted); O = circumcenter(A, B, C); E = rotate(90,B) * (O); F = rotate(90,C) * (O); D = IP(B..E + (B-E)*4, C..F + (C-F)*-3); draw(B--D--C--D--A); P = IP(D..A, A..B..C); dot(A); dot(B); dot(C); dot(D); dot(P); label("$A$", A, dir(335)); label("$B$", B, dir(65)); label("$C$", C, dir(200)); label("$D$", D, dir(135)); label("$P$", P, dir(235)); [/asy]" height="385" src="https://latex.artofproblemsolving.com/1/5/9/15925619d38c6533169fedb4323623e7d6ecb692.png" width="315" /></p>

Question No: 424

Let $\triangle ABC$ have side lengths $AB=5$ , $BC=9$ , $CA=10$ . The tangents to circumcircle of $\triangle ABC$ at $B$ and $C$ intersect at point $D$ , and $\overline{AD}$ intersects the circumcircle at $P \neq A$ . The length of $AP$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers. Find $m + n$ .

Diagram

$[asy] import olympiad; unitsize(15); pair A, B, C, D, E, F, P, O; C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted); O = circumcenter(A, B, C); E = rotate(90,B) * (O); F = rotate(90,C) * (O); D = IP(B..E + (B-E)*4, C..F + (C-F)*-3); draw(B--D--C--D--A); P = IP(D..A, A..B..C); dot(A); dot(B); dot(C); dot(D); dot(P); label("$A$", A, dir(335)); label("$B$", B, dir(65)); label("$C$", C, dir(200)); label("$D$", D, dir(135)); label("$P$", P, dir(235)); [/asy]$

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Tags: Geometry

Question Details

Diagram