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 December 19th, 2018, 08:50 PM #1 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 triangle sorting Why any right-angle triangle cannot be also equilateral triangle? How can I explain it?
December 19th, 2018, 09:24 PM   #2
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Quote:
 Originally Posted by shaharhada Why any right-angle triangle cannot be also equilateral triangle? How can I explain it?
C'mon! You can look up the definitions.

A right triangle is one in which one of the angles is 90 degrees.

An equilateral triangles contains three 60 degree angles.

-Dan

December 19th, 2018, 11:06 PM   #3
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Quote:
 Originally Posted by topsquark C'mon! You can look up the definitions. A right triangle is one in which one of the angles is 90 degrees. An equilateral triangles contains three 60 degree angles. -Dan
Of course I agree, but that's not really the definition of an equilateral triangle.

In fact, in Euclidean geometry as Euclid did it, he never mentioned angles at all except the right angle. So if you wanted to give a proof in the style of Euclid you wouldn't be able to do this.

December 20th, 2018, 03:50 AM   #4
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Quote:
 Originally Posted by shaharhada Why any right-angle triangle cannot be also equilateral triangle?
An equilateral spherical triangle can have three right angles.

December 20th, 2018, 04:29 AM   #5
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 Originally Posted by skipjack An equilateral spherical triangle can have three right angles.
I'll stick with Euclid for now.

Quote:
 Originally Posted by Micrm@ss Of course I agree, but that's not really the definition of an equilateral triangle. In fact, in Euclidean geometry as Euclid did it, he never mentioned angles at all except the right angle. So if you wanted to give a proof in the style of Euclid you wouldn't be able to do this.
It's pretty simple, at least to my Physics mind. Three equal sides means we have a 3-fold axis of symmetry so all the angles have to be equal. And since (skipjack notwithstanding) the sum of the interior angles of a triangle must be "the measure of two right angles" each angle must be 60 degrees. No right angles present.

I hope you don't want me to prove that all right angles have the same measure? I'd have to look that one up.

-Dan

Last edited by topsquark; December 20th, 2018 at 04:32 AM.

December 20th, 2018, 04:34 AM   #6
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Quote:
 Originally Posted by topsquark I'll stick with Euclid for now. It's pretty simple, at least to my Physics mind. Three equal sides means we have a 3-fold axis of symmetry so all the angles have to be equal. And since (skipjack notwithstanding) the sum of the interior angles of a triangle must be 180 degrees each angle must be 60 degrees. No right angles present.
Yes, but never in the elements does Euclid actually measure angles. So something having 60 degrees is alien to him.
I'm not saying your argument is bad, it's perfect for the OP! I'm just wondering about what it would take if you do it perfectly along the philosophy of the elements.

Quote:
 Originally Posted by topsquark I hope you don't want me to prove that all right angles have the same measure? I'd have to look that one up.
That's one of the Euclidean axioms.

Last edited by skipjack; December 20th, 2018 at 06:30 AM.

December 20th, 2018, 04:39 AM   #7
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Quote:
Originally Posted by Micrm@ss
Quote:
 Originally Posted by topsquark I hope you don't want me to prove that all right angles have the same measure?
That's one of the Euclidean axioms.
Just covering myself for skipjack since he wants to play with differential geometry.

-Dan

Last edited by skipjack; December 20th, 2018 at 03:00 PM.

 December 20th, 2018, 07:01 AM #8 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 If I want to explain it to a kindergarten children. How I can do it? By simple way? Any ideas?
December 20th, 2018, 01:21 PM   #9
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Quote:
 Originally Posted by shaharhada If I want to explain it to a kindergarten children. How I can do it? By simple way? Any ideas?
Pictures comparing the two.

December 23rd, 2018, 09:37 AM   #10
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Quote:
 Originally Posted by Micrm@ss So if you wanted to give a proof in the style of Euclid you wouldn't be able to do this.
Euclid's style was hardly flawless.

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