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Previous Puzzles of the Month + Solutions

 
 
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Send to a Friend Puzzle # 131

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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.

Difficulty level: bulbbulb, general geometry knowledge.
Category: Geometry.
Keywords: triangle partition.
Related puzzles:
- Ratio of similar triangles,
- Prof Gibbus Angles.


Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
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Solution

Solution to puzzle 131

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.


cup winnerThe 5 Winners of the Puzzle of the Month are:
Luca Battistella, Italy UK flag - Omar A. Alrefaie, Saudi Arabia Belgium flag - Arthur Vause, UK Dominican Republic flag - Ştefan Gaţachiu, Romania Portugal flag - Luirard Emeline, France Canada flag

Congratulations!

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