MathDB
is (AB+BC+CA)/AD^2 irrational?

Source: China Northern MO 2015 grade 11 p2 CNMO

October 28, 2022
geometryirrational

Problem Statement

As shown in figure , a circle of radius 11 passes through vertex AA of △ABC\vartriangle ABC and is tangent to the side BCBC at the point DD , intersect sides ABAB and ACAC at points EE and FF respectively . AlsoEF EF bisects ∠AFD\angle AFD, and ∠ADC=80o\angle ADC = 80^o , Is there a triangle that satisfies the condition, so that AB+BC+CAAD2\frac{AB+BC+CA}{AD^2} 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. https://cdn.artofproblemsolving.com/attachments/b/9/9e3b955b6d6df35832dd0c0a2d1d2a1e1cce94.png
Back to ProblemsView on AoPS