Problem

Source: China Northern MO 2015 grade 11 p2 CNMO

Tags: geometry, irrational



As shown in figure , a circle of radius 1 passes through vertex A of ABC and is tangent to the side BC at the point D , intersect sides AB and AC at points E and F respectively . AlsoEF bisects AFD, and ADC=80o , Is there a triangle that satisfies the condition, so that AB+BC+CAAD2 is an irrational number, and the irrational number is the root of a quadratic equation with integral coefficients? If it does not exist, please prove it; if it exists, find the quadratic equation that satisfies the condition.