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Geometry Level 3

I'm a fugitive quadrilateral and not a regular polygon. I can always be surrounded by a circumcircle, but I can never surround any inscribed circle.

What non-regular quadrilateral am I?

Kite Rhombus Trapezoid Rectangle Square Parallelogram

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Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

The diagonals of a rectangle act as the hypotenuse of the right triangles, so both diagonals will also be the diameter of a circumcircle. However, a rectangle can't have any inscribed circle because the largest circle that can be inscribed inside will have a diameter of the shorter side of a rectangle. Thus, there can be at most 3 tangents touching the circle, leaving 1 side apart from the circle. Hence, no such inscribed circles exist in a rectangle.

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Creative problem! Just one thing tho.

A rectangle can be a square, and a square can have an inscribed circle

Definition for a rectangle (http://www.merriam-webster.com/dictionary/rectangle): "a four-sided shape that is made up of two pairs of parallel lines and that has four right angles; especially : a shape in which one pair of lines is longer than the other pair"

I would clarify what you mean when you say 'rectangle'.

Thanks.

Vladimir Smith - 5 years, 5 months ago

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How's the edited version?

Worranat Pakornrat - 5 years, 5 months ago

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Great! That fixes it, thanks for the edit.

You missed a spot tho, I would now remove the option 'square', as one of the answers as a square is a regular polygon and could cause confusion. Unless the confusion is an intended aspect you wished to add to your problem :)

Once again, thanks.

Vladimir Smith - 5 years, 5 months ago

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