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Trapezium and circle inscribed in rhombus

Source: Cyprus 2021Junior TST-2 Problem 3

May 26, 2021
geometryrhombustrapezium

Problem Statement

Let ABΓΔAB\varGamma\varDelta be a rhombus.
(a) Prove that you can construct a circle (c)(c) which is inscribed in the rhombus and is tangent to its sides. (b) The points Θ,H,K,I\varTheta,H,K,I are on the sides ΔΓ,BΓ,AB,AΔ\varDelta\varGamma,B\varGamma,AB,A\varDelta of the rhombus respectively, such that the line segments KHKH and IΘI\varTheta are tangent on the circle (c)(c). Prove that the quadrilateral defined from the points Θ,H,K,I\varTheta,H,K,I is a trapezium.
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