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Point Inside an Octagon

Source: 2019 AIME II #13

March 22, 2019
AMCAIMEAIME IIgeometry

Problem Statement

Regular octagon A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 is inscribed in a circle of area 11. Point PP lies inside the circle so that the region bounded by PA1‾\overline{PA_1}, PA2‾\overline{PA_2}, and the minor arc A1A2^\widehat{A_1A_2} of the circle has area 17\tfrac17, while the region bounded by PA3‾\overline{PA_3}, PA4‾\overline{PA_4}, and the minor arc A3A4^\widehat{A_3A_4} of the circle has area 19\tfrac 19. There is a positive integer nn such that the area of the region bounded by PA6‾\overline{PA_6}, PA7‾\overline{PA_7}, and the minor arc A6A7^\widehat{A_6A_7} is equal to 18−2n\tfrac18 - \tfrac{\sqrt 2}n. Find nn.
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