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isosceles wanted, semicircle and circle tangent to arc

Source: 2012 Romania JBMO TST3 P2

May 29, 2020
geometryisoscelestangent circlessemicircle

Problem Statement

Consider a semicircle of center OO and diameter [AB][AB], and let CC be an arbitrary point on the segment (OB)(OB). The perpendicular to the line ABAB through CC intersects the semicircle in DD. A circle centered in PP is tangent to the arc BDBD in FF and to the segments [AB][AB] and [CD][CD] in GG and EE, respectively. Prove that the triangle ADGADG is isosceles.
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