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Let the squares square ABCD and square AB^'C^'D^' share a common polygon vertex A. The midpoints Q and S of the segments B^'D and BD^' together with the centers of the ...

For a point P inside an equilateral triangle DeltaABC, the sum of the perpendiculars p_i from P to the sides of the triangle is equal to the altitude h. This result is simply ...

Tarski's theorem says that the first-order theory of reals with +, *, =, and > allows quantifier elimination. Algorithmic quantifier elimination implies decidability assuming ...

The locus of the centers of all circumconics that also pass through the orthocenter of a triangle (which, when not degenerate, are rectangular hyperbolas) is a circle through ...

A square number, also called a perfect square, is a figurate number of the form S_n=n^2, where n is an integer. The square numbers for n=0, 1, ... are 0, 1, 4, 9, 16, 25, 36, ...

For all n, there exists a k such that the kth term of the Goodstein sequence G_k(n)=0. In other words, every Goodstein sequence converges to 0. The secret underlying ...

The Sobolev embedding theorem is a result in functional analysis which proves that certain Sobolev spaces W^(k,p)(Omega) can be embedded in various spaces including ...

Let A and B_j be sets. Conditional probability requires that P(A intersection B_j)=P(A)P(B_j|A), (1) where intersection denotes intersection ("and"), and also that P(A ...

The four medians of a tetrahedron concur in a point which divides each tetrahedron median in the ratio 1:3, the longer segment being on the side of the vertex of the ...

Let H be a Hilbert space and M a closed subspace of H. Corresponding to any vector x in H, there is a unique vector m_0 in M such that |x-m_0|<=|x-m| for all m in M. ...