perplexus.info :: Geometry : 3 equilateral triangles in a rectangle

3 equilateral triangles in a rectangle (Posted on 2020-02-28)

Let ABCD be a rectangle with DA=12.
ABJ is an equilateral triangle with J inside ABCD.
CHI and EGF are congruent equilateral triangles with E and F on DA, H on CB, J on FG, and E, I, G, H collinear.

Find EF.

Submitted by Jer

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Solution: (Hide)

Let M be the midpoint of AB and the perpendicular bisector of AB intersects EH at point K.
Let N be the midpoint of CH and the perpendicular bisector of CH intersects JM at point L.
Call CH=x and KL=a. CN=x/2. JK=2a.
IL=a*sqrt(3), IN=x*sqrt(3)/2, so LN=BM= x*sqrt(3)/2-a*sqrt(3).
JM=BM*sqrt(3)=3x/2-3a.
ML=BN=3x/2.
CB=2x.
Since CB=DA=12, EF=CH=6.
Note: There is a nice invariant here. The solution does not vary with the size of the congruent triangles.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: My attempt	Jer	2020-03-14 17:39:47
My attempt	tomarken	2020-02-28 11:02:16

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