This question is related to a component of a proof I constructed. I have posted the (pending) proof for verification on MSE: https://math.stackexchange.com/questions/2067151/use-the-information-given-to-construct-a-proof.

However, I would greatly appreciate it if someone could verify whether this specific statement I made is true:

If $XYZ$ is a right scalene triangle and if one of its constituent triangles is equilateral, then the other triangle must be isosceles (and vice-versa).

In the above proof, this statement was made to justify the following key question and abstract answer:

How can I show that a right scalene triangle consists of an equilateral triangle and an isosceles triangle?

Show that one of the triangles is an equilateral triangle.

In the above question, I also included the following diagram to illustrate the triangles:

enter image description here

Notice that the right scalene triangle consists of two other triangles when divided.

If anything is unclear, you may find clarification in the proof question posted above. Also, I will be glad to answer any questions if you'd prefer to ask me directly.

If my reasoning is incorrect, please explain why.

Thank you.


closed as unclear what you're asking by Jean Marie, Rohan, Henrik, E. Joseph, John B Dec 23 '16 at 9:25

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  • $\begingroup$ What is $W$ here\? $\endgroup$ – Qwerty Dec 23 '16 at 6:51
  • $\begingroup$ @Qwerty $W$ is the midpoint of the hypotenuse. $\endgroup$ – The Pointer Dec 23 '16 at 6:52
  • $\begingroup$ What does bring the adjective "scalene" in the title ? Do you mean by this "any right triangle" ? $\endgroup$ – Jean Marie Dec 23 '16 at 7:01
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    $\begingroup$ @ThePointer In any right triangle the midpoint of the hypothenuse is the center of the circumscribed circle, so $WX = WY \implies WX = WY=WZ$. This was addressed in the original question and answer. $\endgroup$ – dxiv Dec 23 '16 at 7:06
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    $\begingroup$ You didn't really include any reasoning in this posted question. if one of its constituent triangles is equilateral, then the other triangle must be isosceles (and vice-versa) The vice-versa part is obviously false, take an isosceles right triangle for example. $\endgroup$ – dxiv Dec 23 '16 at 7:12

First question. If the constituent triange is equilateral then one of the angels of the right triangle is 60. And the other is 30. Removing the equilateral triangle you'll be left with a triangle with the 30 degree angle and the 90 degree angle less 60 degrees. As two of the angles are both 30, it is isoceles.

You second question is clearly false as only 60-30-90 triangles will have this property.

  • $\begingroup$ Thank you for the answer. Which second question are you referring to? This is only the context of the 60-30-90 triangle. $\endgroup$ – The Pointer Dec 23 '16 at 7:16
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    $\begingroup$ "How can I show that a right scale ne triangle does...?" That second question. It isn't true that right scales have this property. 60-30-90s are the only ones that do. So the property is false in general. $\endgroup$ – fleablood Dec 23 '16 at 7:19
  • $\begingroup$ Ahh, you're correct. In that case, this error invalidates my proof, since it assumes the hypothesis true in constructing the conclusion. Thank you for the assistance. $\endgroup$ – The Pointer Dec 23 '16 at 7:21

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