query: Prove that $\rm area(\triangle ABE) = area (ABCD)$
positive_passages.docid: 153277
positive_passages.text: When is $Ar(APD)=Ar(ABCD)$?
negative_passages.docid: 46106
negative_passages.text: Prove that $D_r \circ D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y|<r+s \}$, where $D_a = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < a \}$
negative_passages.docid: 57406
negative_passages.text: Proving the volume of a cone
negative_passages.docid: 93599
negative_passages.text: General Proof for the triangle inequality
negative_passages.docid: 134652
negative_passages.text: Prove that the perpendicular bisectors of the sides of a triangle meet at a point
negative_passages.docid: 60795
negative_passages.text: Quadrilateral's area problem
negative_passages.docid: 81794
negative_passages.text: Prove that $(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$
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: 104217
negative_passages.text: Alternative triangle inequality proof
negative_passages.docid: 216
negative_passages.text: In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is
negative_passages.docid: 123465
negative_passages.text: In triangle $ABC$, $AA_1$, $BB_1$, $CC_1$ divide sides in ratio of $1: 2$ and meet at $M$, $K$, $L$. Find area relation of $KLM$ and $ABC$
negative_passages.docid: 25825
negative_passages.text: Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
negative_passages.docid: 39015
negative_passages.text: Prove the triangle inequality on the sphere $S^2$ in $\mathbb{R}^3$.
negative_passages.docid: 133567
negative_passages.text: Area of a triangle in terms of areas of certain subtriangles
negative_passages.docid: 35589
negative_passages.text: Exact area using limits and Riemann sum
negative_passages.docid: 69203
negative_passages.text: Calculus proof for the area of a circle
negative_passages.docid: 15365
negative_passages.text: Proving that the triangle inequality holds for a metric on $\mathbb{C}$