# Find The Ratio

Geometry Level 4

In acute $$\triangle ABC,$$ let $$D, E, F$$ be the feet of perpendiculars from $$A, B, C$$ to $$BC, CA, AB$$ respectively. Line $$EF$$ meets the circumcircle of $$\triangle ABC$$ at two distinct points $$P$$ and $$Q.$$ $$DF$$ intersects $$BP$$ and $$BQ$$ at points $$R, S$$ respectively. Also, line $$DE$$ intersects $$CQ$$ and $$CP$$ at points $$T, U$$ respectively. Find $$\dfrac{AR+AS+AT}{AP+AQ+AU}.$$