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Browse Classroom Capsules and Notes

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Displaying 1021 - 1030 of 1209

The tangent of the sum formula is proved by using an inscribed quadrilateral whose longest side is the diameter of the circle, and the diagonals are perpendicular to the pair of oposite sides...

The author shows that every Heron triangle is similar to one with sides \(r + 1/r\), \(s + 1/s\), and \(r - 1/r + s -1/s\) for rational numbers \(r\) and \(s\) greater than one.
The author gives geometrical proofs of a number of identities for the Fibonacci numbers.

A 1932 Erdos conjecture on packing squares into a square is shown to be equivalent to a more general conjecture.

The authors give two proofs, one geometric and one algebraic, to provide insight into why the sum of independent normal. random variables must be normal.
The author shows that every triangle can be dissected into four (acute triangles into three) isosceles triangles, and that a triangle can be dissected into two isosceles triangles if and only if...
It is shown that any bijective function that maps circles and lines onto circles and lines is either a linear fractional transformation, or the complex conjugate of a linear fractional...
The author characterizes the set of possible limits of subsequences of a given sequence.
Two visual proofs of some exponential inequalities are presented.
Given any positive integer \(n\), find polynomial curves that have exactly \(n\) tangent lines which intersect the curve exactly once.

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