3

Part of 1982 Vietnam National Olympiad

Problems(2)

Equality of areas for equilateral triangles on sides...

Source: Vietnam MO 1982 P3

Let be given a triangle

ABC

. Equilateral triangles

BCA_1

BCA_2

are drawn so that

A

A_1

are on one side of

BC

A_2

is on the other side. Points

B_1,B_2,C_1,C_2

are analogously defined. Prove that

S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.

geometrygeometry unsolved

No line intersects four lines in a cube ABCDA'B'C'D'

Source: Vietnam MO 1982 P6

ABCDA'B'C'D'

be a cube (where

ABCD

A'B'C'D'

AA',BB',CC',DD'

are edges). Consider the four lines

AA', BC, D'C'

and the line joining the midpoints of

BB'

DD'

. Show that there is no line which cuts all the four lines.

geometry3D geometrygeometry unsolved