MathDB

2

Part of 2018 Oral Moscow Geometry Olympiad

Problems(2)

trapezoid, angle bisectors and equidistant midpoint

Source: 2018 Oral Moscow Geometry Olympiad grades 8-9 p2

7/25/2019
Bisectors of angle CC and externalangle AA of trapezoid ABCDABCD with bases BCBC and ADAD intersect at point MM, and the bisector of angle BB and external angle DD intersect at point NN. Prove that the midpoint of the segment MNMN is equidistant from the lines ABAB and CDCD.
geometrytrapezoidangle bisectormidpoint
right angle wanted, starting with a trapezoid with perpendicular diagonals

Source: 2018 Oral Moscow Geometry Olympiad grades 10-11 p2

7/25/2019
The diagonals of the trapezoid ABCDABCD are perpendicular (AD//BC,AD>BCAD//BC, AD>BC) . Point MM is the midpoint of the side of ABAB, the point NN is symmetric of the center of the circumscribed circle of the triangle ABDABD wrt ADAD. Prove that ∠CMN=90o\angle CMN = 90^o.
(A. Mudgal, India)
geometrytrapezoidperpendiculardiagonalsright angle