# A geometry problem by Danish Ahmed

Geometry Level 3

In triangle $$PQR$$, $$PQ = QR$$ and $$RS$$ is the altitude. $$PR$$ is extended to point $$T$$ such that $$QT = 10$$.

The values of $$\tan \angle RQT, \tan \angle SQT$$ and $$\tan \angle PQT$$ form a geometric sequence, and:

The values of $$\cot \angle SQT, \cot \angle RQT$$ and $$\cot \angle SQR$$ form an arithmetic sequence.

If the area of the triangle $$PQR$$ can be exprssed as $$\dfrac{a}{b}$$ where $$gcd(a ,b) = 1$$ .

Then $$a + b =$$

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