MathDB
2018 Geometry #6

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February 12, 2018
geometry

Problem Statement

Let ABCABC be an equilateral triangle of side length 1.1. For a real number 0<x<0.5,0<x<0.5, let A1A_1 and A2A_2 be the points on side BCBC such that A1B=A2C=x,A_1B=A_2C=x, and let TA=△AA1A2.T_A=\triangle AA_1A_2. Construct triangles TB=△BB1B2T_B=\triangle BB_1B_2 and TC=△CC1C2T_C=\triangle CC_1C_2 similarly.
There exist positive rational numbers b,cb,c such that the region of points inside all three triangles TA,TB,TCT_A,T_B,T_C is a hexagon with area 8x2−bx+c(2−x)(x+1)⋅34.\dfrac{8x^2-bx+c}{(2-x)(x+1)}\cdot \dfrac{\sqrt 3}{4}. Find (b,c).(b,c).
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