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Geometry Level 4

Suppose that A B \overline{AB} and C D \overline{CD} intersect at X X , and that A X = 6 AX=6 , X B = 4 XB=4 , C X = 3 CX=3 , and X D = 8 XD=8 . Is quadrilateral A B C D ABCD cyclic? (The quadrilateral is non-degenerate).

Yes Not enough Information No

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1 solution

Alex Li
Jun 23, 2015

Normally, by the converse of Power of a Point, it would be cyclic. However, it is possible that the quadrilateral is concave, which obviously makes it non-cyclic. For example, we could have C > 18 0 \angle C>180^\circ , and A B \overline{AB} and C D \overline{CD} intersect on side A B AB . Therefore, we cannot conclusively determine whether the quadrilateral is cyclic.

Moderator note:

For clarity, you should ensure that you are using proper notation. Very often, A B \overline{AB} is used to refer to the line segment A B AB , while A B \overleftrightarrow{AB} is used to refer to the line A B AB that extends in both directions.

In the former case, we are guaranteed that these points are concyclic.

Nice explanation. I didn't think about that case.

Anupam Nayak - 5 years, 4 months ago

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