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gamma1,2,3 are semicircles show that AB=DE

Source: India tst 2002 p1

June 30, 2012
geometrytrapezoidgeometric transformationhomothetypower of a pointradical axisgeometry proposed

Problem Statement

Let A,BA,B and CC be three points on a line with BB between AA and CC. Let Γ1,Γ2,Γ3\Gamma_1,\Gamma_2, \Gamma_3 be semicircles, all on the same side of ACAC and with AC,AB,BCAC,AB,BC as diameters, respectively. Let ll be the line perpendicular to ACAC through BB. Let Γ\Gamma be the circle which is tangent to the line ll, tangent to Γ1\Gamma_1 internally, and tangent to Γ3\Gamma_3 externally. Let DD be the point of contact of Γ\Gamma and Γ3\Gamma_3. The diameter of Γ\Gamma through DD meets ll in EE. Show that AB=DEAB=DE.
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