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Intermediate

Ménage à trois

Three equilateral triangles of which two are congruent. Are the three points collinear?

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Solution

Yes, the three points are collinear.

The first step is to construct the circle which is centred at the common vertex of the small triangles and has their side as radius. Now obviously the angle EAD is 60°.

Next the angle DBF is 30°, because BD is the bisector of 60°. Also, from the Inscribed angle theorem we infer that angle DBE is 30°. Thus, B, F and E must lie on a line.

Poem

Today is possible some polyandrie
In an conformist society
When you can still find polygamie
May be 3 figures in geometry
Having fun with one another
We think immediately for 3
The souls singing and dancing
And 3 broken Hearts
End of 3 loves,
Sadness
Melancholy

Bella

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