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Problem 5

Part of 1972 Bulgaria National Olympiad

Problems(1)

quadrilateral with _|_ diagonals, perpendiculars from intersection point

Source: Bulgaria 1972 P5

6/21/2021
In a circle with radius RR, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.
(a) Prove that the feet of these perpendiculars P1,P2,P3,P4P_1,P_2,P_3,P_4 are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities 2r1≤2R1≤R2r_1\le\sqrt2 R_1\le R where R1R_1 and r1r_1 are radii respectively of the circumcircle and inscircle to the quadrilateral P1P2P3P4P_1P_2P_3P_4. When does equality hold?
H. Lesov
geometrycyclic quadrilateralperpendicularinequalitiesGeometric Inequalities