Monthly Archives: January 2016

Triangles with 60° angle and sides integers

  Previous blog:   Integer triangles with 120° angle   Law of cosines https://en.wikipedia.org/wiki/Law_of_cosines If     —>   then,         To find triples     that satisfy the relation   We use the parametric equations where   … Continue reading

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Integer Triangles such the median & Altitude to/from the largest side are all integers

    To find triangles     such that its sides, the median   (     )   to the largest side   and the altitude from     are all integers.   In     triangle, the lengths … Continue reading

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Integer Triangles (a,b,c) such that the median to the largest side is an integer

    To find integer triangles     such that the median   (     )   to the largest side is an integer.   In     triangle, the lengths of the medians :   For example,   … Continue reading

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Some solutions to Project Euler 62: Cubic permutations

  Project Euler 62 https://projecteuler.net/index.php?section=problems&id=62 The cube,   ,   can be permuted to produce two other cubes:    and In fact,   41063625   is the smallest cube which has exactly three permutations of its digits which are also … Continue reading

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Integer triangles with 120° angle

  Law of cosines https://en.wikipedia.org/wiki/Law_of_cosines If     —>   then,         To find triples     that satisfy the relation   We use the parametric equations : where     and     are integers and   … Continue reading

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Triangle (A,B,C) such that tan A, tan B, and tan C integers?

    For any triangle ABC, the proof can be easily derived from simple trigonometric identities.     Is it possible for all three of   ,   ,   and     to be integers? If so,   how … Continue reading

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(x,y,z) positive integers, √x + √y = √z, z ≤ 1000

    ,   and     are positive integers such that     Find all possible values of   A pattern will emerge.   Find it.     Solution:     must be a square. In general,   , … Continue reading

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Primitive Heron triangles – two sides have the common factor 5

    The primitive Heron triangle   (45, 296, 325)   in which two sides have the common factor 5. area   =   5328 perimeter   =   666   Note that 325 is precisely the length of the … Continue reading

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Heron Triangle with square sides

  Heron Triangle with square sides   Area   =   32918611718880 Perimeter   =   42216338     has 6 representations as a sum of 2 squares:                         … Continue reading

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Heron triangles (17, n, n+3), (17, n, n+5) and (17, n, n+1)

  Older post:   Heron triangles (17, n, n+3) and (17, n, n+5)   Here are the first few examples:                                

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