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# First 2014 puzzle. Dissection of a triangle

10 replies to this topic

### #1 bonanova

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Posted 01 January 2014 - 06:00 AM

Place an arbitrary point P inside an isosceles equilateral triangle.
From P, draw lines to the vertices and perpendiculars to the sides.
Six smaller triangles now touch at P.

Prove that the sums of areas of the two alternating sets of three triangles are equal.

Edited by bonanova, 01 January 2014 - 11:16 PM.
Correction

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### #2 plasmid

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Posted 01 January 2014 - 04:23 PM

Do you mean equilateral instead of isosceles?
Spoiler for counterexample for isosceles

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### #3 bonanova

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Posted 01 January 2014 - 10:57 PM

Do you mean equilateral instead of isosceles?

Spoiler for counterexample for isosceles

Quite right. Equilateral. I changed the OP.
Thanks.

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### #4 plasmid

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Posted 22 June 2014 - 04:21 PM

Spoiler for

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### #5 plasmid

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Posted 22 June 2014 - 04:55 PM

Drat. I just looked over that again and saw that I divided instead of multiplying when manipulating that equation after introducing the deltaA term, so it won't work.

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### #6 plasmid

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Posted 22 June 2014 - 10:09 PM

Spoiler for revised answer with changes in red

Edited by plasmid, 23 June 2014 - 03:05 PM.

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### #7 plainglazed

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Posted 23 June 2014 - 03:44 PM

Spoiler for maybe

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### #8 k-man

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Posted 23 June 2014 - 05:03 PM   Best Answer

Spoiler for Aha!

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### #9 plainglazed

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Posted 23 June 2014 - 05:14 PM

très élégant k-man!

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### #10 plasmid

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Posted 24 June 2014 - 07:13 AM

Indeed, beautiful solution, k-man!
(Especially compared to all that slogging I was doing. )
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