query: If opposite angles in a quadrilateral add up to π, do its four vertices all lie on the same circle
positive_passages.docid: 25079
positive_passages.text: Prove the opposite angles of a quadrilateral are supplementary implies it is cyclic.
negative_passages.docid: 134805
negative_passages.text: How many sides does a circle have?
negative_passages.docid: 74505
negative_passages.text: 3 square in a circle (geometric question)
negative_passages.docid: 2786
negative_passages.text: The staircase paradox, or why $\pi\ne4$
negative_passages.docid: 155459
negative_passages.text: How to find out the solid angle subtended by a tetrahedron at its vertex?
negative_passages.docid: 126040
negative_passages.text: Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$
negative_passages.docid: 58953
negative_passages.text: Area of intersection between 4 circles centered at the vertices of a square
negative_passages.docid: 70183
negative_passages.text: Angular Momentum Addition Theorem - Sanity Check
negative_passages.docid: 118501
negative_passages.text: Proving the length of a circle's arc is proportional to the size of the angle
negative_passages.docid: 57274
negative_passages.text: Asphericity of pseudo circle
negative_passages.docid: 45356
negative_passages.text: Are there any positive integers $a, b, c, d$ such that both $(a, b, c)$ and $(b, c, d)$ are Pythagorean triples?
negative_passages.docid: 61344
negative_passages.text: How to relate areas of circle, square, rectangle and triangle if they have same perimeter??
negative_passages.docid: 74574
negative_passages.text: Are all equiangular odd polygons also equilateral?
negative_passages.docid: 63668
negative_passages.text: Is the 1-dimension projective line a circle?
negative_passages.docid: 15003
negative_passages.text: Pythagorean Quadruples:
negative_passages.docid: 118716
negative_passages.text: If $ABCD$ is a cyclic quadrilateral, then $AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)$
negative_passages.docid: 138236
negative_passages.text: What is the sum of the angles of a triangle on Earth orbit?