MathDB

BC\AB =BB' /BC+ BC/BB', planes angles, cube related (2018 Romanian NMO VIII P4)

Source:

June 3, 2020

geometry3D geometrycubeanglesparallelequal segments

Problem Statement

In the rectangular parallelepiped

ABCDA'B'C'D'

we denote by

M

the center of the face

ABB'A'

. We denote by

M_1

and

M_2

the projections of

M

on the lines

B'C

and

AD'

respectively. Prove that:
a)

MM_1 = MM_2

b) if

(MM_1M_2) \cap (ABC) = d

, then

d \parallel AD

;
c)

\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}