Equal
partitions ABC is
an equilateral triangle. G is an arbitrary point
inside the triangle. Segments GD, GE and GF are
perpendiculars to the sides. Prove that the areas k + m + o and l + n + p are
equal.
- Spartizioni All'interno del triangolo equilatero ABC viene
messo in modo arbitrario un punto G. I segmenti GD, GE e GF sono
perpendicolari ai lati del triangolo. Dimostra che
le aree k + m + o e l + n + p sono
uguali.
- Partages équitaux On place de façon arbitraire un point G à l'intérieur
du triangle équilatéral ABC.
Les segmensi GD, GE et GF sont
perpendiculaires aux côtés du triangle.
Démontrez que k + m + o et l + n + p ont
la même aire.
Draw
3 additional lines through point G,
each parallel to a side of the main triangle divinding
it into 3 parallelograms and 3 equilateral triangles.
The parallelograms are bisected by their diagonals
and the triangles by their height, so partitioned
areas: k’ = l’’; l’ = m’’;
m’ = n’’; n’ = o’’;
o’ = p’’; and p’ = k’’
Thus: k + m + n =
k’’ + k’ + m’’ + m’ +
o’’ + o’ =
= p’ + l’’ + l’ + n’’ +
n’ + p’’ = p + n + l Q.E.D.
The
5 Winners of the Puzzle of the Month are: Luca Battistella, Italy - Omar
A. Alrefaie, Saudi Arabia - Arthur
Vause, UK - Ştefan
Gaţachiu, Romania - Luirard
Emeline, France