MathDB
Tangent Tangents

Source: Brazil National Olympiad Junior 2021 #7

February 15, 2022
geometry

Problem Statement

Let ABCABC be a triangle with ∠ABC=90∘\angle ABC=90^{\circ}. The square BDEFBDEF is inscribed in △ABC\triangle ABC, such that D,E,FD,E,F are in the sides AB,CA,BCAB,CA,BC respectively. The inradius of △EFC\triangle EFC and △EDA\triangle EDA are cc and bb, respectively. Four circles ω1,ω2,ω3,ω4\omega_1,\omega_2,\omega_3,\omega_4 are drawn inside the square BDEFBDEF, such that the radius of ω1\omega_1 and ω3\omega_3 are both equal to bb and the radius of ω2\omega_2 and ω4\omega_4 are both equal to aa. The circle ω1\omega_1 is tangent to EDED, the circle ω3\omega_3 is tangent to BFBF, ω2\omega_2 is tangent to EFEF and ω4\omega_4 is tangent to BDBD, each one of these circles are tangent to the two closest circles and the circles ω1\omega_1 and ω3\omega_3 are tangents. Determine the ratio ca\frac{c}{a}.
Back to ProblemsView on AoPS