2019 AIME I Problems/Problem 6
In convex quadrilateral side is perpendicular to diagonal , side is perpendicular to diagonal , , and . The line through perpendicular to side intersects diagonal at with . Find .
Solution 1 (Simplicity)
Note that is cyclic with diameter since . Also, note that we have by SS similarity.
We see this by and . The latter equality can be seen if we extend to point on . We know from which it follows .
Let . By we have
Solution 2 (Trig)
Let and . Note .
Then, . Furthermore, .
Dividing the equations gives
Thus, , so .
Solution 3 (Similar triangles)
First, let be the intersection of and as shown above. Note that as given in the problem. Since and , by AA similarity. Similarly, . Using these similarities we see that and Combining the two equations, we get Since , we get .
Solution by vedadehhc
Solution 4 (Similar triangles, orthocenters)
Extend and past and respectively to meet at . Let be the intersection of diagonals and (this is the orthocenter of ).
As (as , using the fact that is the orthocenter), we may let and .
Then using similarity with triangles and we have
Cross-multiplying and dividing by gives so . (Solution by scrabbler94)
Solution 5 (5-second PoP)
Notice that is inscribed in the circle with diameter and is inscribed in the circle with diameter . Furthermore, is tangent to . Then, and .
(Solution by TheUltimate123)
Video Solution: https://www.youtube.com/watch?v=0AXF-5SsLc8
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