Justifying Equilateral Triangle AEF in ABCD

In summary: The thread will be closed for now.In summary, the problem states that there is a rectangle ABCD with points E and F such that triangles BEC and CFD are equilateral and share only one side with the rectangle. The question is then how to prove that triangle AEF is also equilateral. The key observation is that an equilateral triangle has all angles of 60 degrees, so if we can show that the angles in AEF are all 60 degrees, then it must be equilateral as well.
  • #1
charlie05
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Poster warned (yet again) to always show their work when posting schoolwork questions
Homework Statement
see attached, how do I prove? When drawn, a triangle has all angles of 60 degrees, so it is equilateral. Thank you.
Relevant Equations
an equilateral triangle has all angles 60 degrees
Given a rectangle ABCD and points E, F such that the triangles BEC and CF D are equilateral and each of them has only one side in common with the rectangle ABCD. Justify that triangle AEF is also equilateral.
 
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  • #2
charlie05 said:
Homework Statement:: see attached, how do I prove? When drawn, a triangle has all angles of 60 degrees, so it is equilateral. Thank you.
Relevant Equations:: an equilateral triangle has all angles 60 degrees

Given a rectangle ABCD and points E, F such that the triangles BEC and CF D are equilateral and each of them has only one side in common with the rectangle ABCD. Justify that triangle AEF is also equilateral.
What attachment?

-Dan
 
  • #3
by attachment I mean the task text listed below under the heading, there is no other attachment, sorry for the mistake
 
  • #4
Hi. In-line with forum rules, you need to show us evidence of your own effort. A labelled diagram with the angles you can work-out would probably be a bare minimum.

Edit: you might want to check that you have stated the question completely and accurately, as it looks wrong to me.
Edit: sorry - the question look ok.
 
Last edited:
  • #5
the full text of the task reads: : a rectangle ABCD and points E, F such that triangles BEC and CFD are equilateral and each of them has only one side in common with the rectangle ABCD. Reason that triangle AEF is also equilateral.
 

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  • #6
charlie05 said:
Homework Statement:: see attached, how do I prove? When drawn, a triangle has all angles of 60 degrees, so it is equilateral. Thank you.
Relevant Equations:: an equilateral triangle has all angles 60 degrees

Given a rectangle ABCD and points E, F such that the triangles BEC and CF D are equilateral and each of them has only one side in common with the rectangle ABCD. Justify that triangle AEF is also equilateral.
Thread is now closed. @charlie05 -- You have been reminded multiple times in the past that you MUST show your best efforts to work on schoolwork problems when posting at PF. Please start a new thread on this question and show your work to try to answer it.
 
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FAQ: Justifying Equilateral Triangle AEF in ABCD

What is an equilateral triangle?

An equilateral triangle is a triangle in which all three sides are of equal length and all three interior angles are equal, each measuring 60 degrees.

How do you identify points E and F in triangle AEF within quadrilateral ABCD?

Points E and F are typically identified by specific geometric constructions or conditions given in the problem. For example, they could be points on sides AB and AD of quadrilateral ABCD such that AE = EF = AF, forming an equilateral triangle AEF.

What are the necessary conditions for triangle AEF to be equilateral in quadrilateral ABCD?

The necessary conditions for triangle AEF to be equilateral within quadrilateral ABCD include ensuring that AE = EF = AF. This can be achieved through geometric constructions or by setting specific lengths and angles within the quadrilateral.

How can you prove that triangle AEF is equilateral?

To prove that triangle AEF is equilateral, you need to show that all three sides AE, EF, and AF are equal. This can involve using properties of the quadrilateral, congruence criteria, and geometric constructions to demonstrate the equality of the sides and the angles.

What role do the properties of quadrilateral ABCD play in justifying triangle AEF as equilateral?

The properties of quadrilateral ABCD, such as the lengths of its sides, its angles, and the relationships between its diagonals, can provide the necessary information to construct points E and F and to verify that AE = EF = AF. These properties help in forming the geometric basis for proving that triangle AEF is equilateral.

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