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Beauty and the Bash

Source: 2022 AIME I #14

February 9, 2022
AMCAIMEAIME I

Problem Statement

Given △ABC\triangle ABC and a point PP on one of its sides, call line ℓ\ell the splitting line of △ABC\triangle ABC through PP if ℓ\ell passes through PP and divides △ABC\triangle ABC into two polygons of equal perimeter. Let △ABC\triangle ABC be a triangle where BC=219BC = 219 and ABAB and ACAC are positive integers. Let MM and NN be the midpoints of AB‾\overline{AB} and AC‾\overline{AC}, respectively, and suppose that the splitting lines of △ABC\triangle ABC through MM and NN intersect at 30∘30^{\circ}. Find the perimeter of △ABC\triangle ABC.
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