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Part of 2018 Oral Moscow Geometry Olympiad

Problems(2)

intersecting parallelograms , intersecting diagonals problem

Source: 2018 Oral Moscow Geometry Olympiad grades 8-9 p1

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second. https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png

geometryparallelogramdiagonal

projections distance on angle bisectors equals to inradius

Source: 2018 Oral Moscow Geometry Olympiad grades 10-11 p1

In a right triangle

ABC

with a right angle

C

AK

BN

be the angle bisectors. Let

D,E

be the projections of

C

AK, BN

respectively. Prove that the length of the segment

DE

is equal to the radius of the inscribed circle.

right trianglegeometryinradiusprojections