MathDB

two lines, another _|_ to both in space

Source: Bulgaria 1965 P4

June 23, 2021

geometry3D geometry

Problem Statement

In the space there are given crossed lines

s

and

t

such that

\angle(s,t)=60^\circ

and a segment

AB

perpendicular to them. On

AB

it is chosen a point

C

for which

AC:CB=2:1

and the points

M

and

N

are moving on the lines

s

and

t

in such a way that

AM=2BN

. The angle between vectors

\overrightarrow{AM}

and

\overrightarrow{BM}

is

60^\circ

. Prove that:
(a) the segment

MN

is perpendicular to

t

; (b) the plane

\alpha

, perpendicular to

AB

in point

C

, intersects the plane

CMN

on fixed line

\ell

with given direction in respect to

s

; (c) all planes passing by

ell

and perpendicular to

AB

intersect the lines

s

and

t

respectively at points

M

and

N

for which

AM=2BN

and

MN\perp t

.

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