2020 USAMO Problems/Problem 1
Problem 1
Let be a fixed acute triangle inscribed in a circle with center . A variable point is chosen on minor arc of , and segments and meet at . Denote by and the circumcenters of triangles and , respectively. Determine all points for which the area of triangle is minimized.
Solution
Let be midpoint Let be midpoint and are the bases of perpendiculars dropped from and respectively.
Therefore
is cyclic)
Similarly
The area of is minimized if because $$ (Error compiling LaTeX. Unknown error_msg)\frac {[OO_1O_2]} {[ABC]} = (\frac {O_1 O_2} {AB})^2 \ge (\frac {EF} {AB})^2 = \frac {1}{4}$.