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Triangle ABC has side lengths $A B = 7, B C = 8$ and $C A = 9.$ Circle $ω_{1}$ passes through B and is tangent to line AC at A. Circle $ω_{2}$ passes through C and it tangent to line AB at A. Let K be in the intersection of circles $ω_{1}$ and $ω_{2}$ ( K is different from A). Then $A K = \frac{m}{n},$ where m and n are relatively prime positive integers. The value of $m - n$ is equal to _____

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answer is 7.

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Detailed Solution

Note that from the tangency condition that the supplement of $∠ C A B$ with respects to lines AB and AC are equal to AC and $∠ A K C,$ respectively, so from tangent chord, $∠ A K C = ∠ A K B = 180 ° - ∠ B A C$

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