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# Let ABC be any triangle. Equilateral triangles BCX, ACY, and BAZ are constructed such that none of these triangles overlaps triangle ABC.

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Let ABC be any triangle. Equilateral triangles BCX, ACY, and BAZ are constructed such that none of these triangles overlaps triangle ABC.

a) Draw a triangle ABC and then sketch the remainder of the figure. It will help if triangle ABC is not isosceles (or equilateral).

b) Show that, regardless of choice of triangle ABC, we always have AX = BY = CZ.

You must include a diagram for any complicated geometry problem that does not come with its own diagram. Diagrams are extremely important in geometry because they help your reader understand what is going on. Diagrams also help you to keep track of your work and possibly give you useful insight.

Thanks so much!

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#1
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Here is the pic but I don't have an answer yet.

Melody  Feb 7, 2018
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Chris you are better at these than I am, why don't you give it a go?

Melody  Feb 7, 2018
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Using your diagram, I can prove that  AX  = BY

Note that we have two triangles, ACX  and BCY

AC    =  CY   =  "b"

And

XC  = BC  =  "a"

And  angle ACX  = angle YCB

So.... by SAS   .... triangle  ACX  is congruent to triangle YCB

So... AX  =  BY

But...I have not figured out how to prove that CZ  is also equal to these

Maybe someone can provide the final step  ???

CPhill  Feb 7, 2018
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Huh I guess this was a harder problem than I thought!  Sorry for the impatience, it was just due yesterday so I got really antsy.  Thanks for the help everyone!

#6
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Don't appologise, your repost request was done exactly as I always asked for it to be done.

Melody  Feb 8, 2018
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Thanks Chris,

The bit that follows is exactly the same as Chris said and then i have continued the proof

I am going to call the angles in the original triangle

All angles in the 3 other triangles are all 60 degrees because they are equilateral triangles.

Consider

$$\triangle ACX\;\; and \;\; \triangle YCB \\ AC=b, \;\;CX=a\;\; and \;\; \angle ACX=\angle C+60^\circ\\ YC=b, \;\;CB=a\;\; and \;\; \angle YCB=\angle C+60^\circ\\ \therefore \triangle ACX\cong \triangle YCB\qquad \text{(2 sides and included angle test)} \\ \therefore AX=BY \qquad \text{Corresponding sides in congruent triangles.}$$

$$\text{Now consider} \triangle ABY\;\; and \;\; \triangle AZC \\ AB=AZ=c\\ AY=AC=b\\ \angle YAB=\angle CAZ=\angle A+60^\circ\\ \therefore \triangle ABY\cong \triangle AZC\qquad \text{(2 sides and included angle test)} \\ \therefore BY=ZC \qquad \text{Corresponding sides in congruent triangles.}\\$$

$$\therefore AX=BY=ZC\\~\\ QED$$

Melody  Feb 8, 2018
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OH! Thanks so much!  I can use this actually for another problem that stumped me!

#7
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Ah....thanks, Melody.....it's funny how it's sometimes hard to see....and then when someone else points it out.....it seems really easy.....LOL!!!!!

I think that my drawing of this maybe wasn't quite correct.....Melody's made it easy to see.....

CPhill  Feb 8, 2018

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