Can an isosceles triangle have 3 equal angles?

[Saracen Rue]

Okay, I know I must sound like a complete idiot here, but please bear with me.

I've come across a scenario in which I have triangle ECF. Angle ECF = 60 degrees, Angles CEF and CFE are unknown, lengths EC and FC are unknown and equal and length EF is r√3 and not equal to lengths EC and FC

I know it has to be an isosceles triangle, but because angle ECF = 60 the remaining angles must both also be equal to 60 degrees.

I'm not sure if this is possible or not. Also, sorry if this is a very basic question but I have honestly never encountered this scenario before. Any help is much appreciated.

 

[Mentallic]

Okay, I know I must sound like a complete idiot here, but please bear with me.

I've come across a scenario in which I have triangle ECF. Angle ECF = 60 degrees, Angles CEF and CFE are unknown, lengths EC and FC are unknown and equal and length EF is r√3 and not equal to lengths EC and FC

I know it has to be an isosceles triangle, but because angle ECF = 60 the remaining angles must both also be equal to 60 degrees.

I'm not sure if this is possible or not. Also, sorry if this is a very basic question but I have honestly never encountered this scenario before. Any help is much appreciated.

This is an impossible object.

However, isosceles triangles can have 3 equal angles, and it becomes a special type of isosceles triangle - the equilateral triangle. This is equivalent to a square being a special type of rectangle, which itself is a special type of parallelogram, which is a special quadrilateral.

 

[mfb]

Such a triangle is possible on a sphere and some other objects, but not on a plane.