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All faces of the parallelepiped

ABCDA_1B_1C_1D_1

are equal rhombuses. Plane angles at vertex

A

are equal. Points

K

and

M

are taken on the edges

A_1B_1

and

A_1D_1

. It is known that

A_1K = a

A_1M = b

, and

a + b

is an edge of the parallelepiped. Prove that the plane

AKM

touches the sphere inscribed in the parallelepiped. Let us denote by

Q

the touchpoint of this sphere with the plane

AKM

. In what ratio does the straight line

AQ

divide the segment

KM

Problem Statement