Geometry, Plane Geometry, Triangle Inequality

The Triangle Inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. Questions involve determining if given lengths can form a triangle or finding bounds for a side length.

• Question

  Given a line `l` and two points `A, B` at different distances from the line. Find the point `C` on the line such that the difference between the lengths of the segments `AC`, `AB` is maximal.

  Topics:

  Geometry -> Plane Geometry -> Triangle Inequality Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 3

• Question

  In the plane, a square and a point `P` are given. Prove that it is impossible for the distances from `P` to the vertices of the square to be `1`, `1`, `2`, and `3` centimeters?

  Topics:

  Geometry -> Plane Geometry -> Triangles Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality

• Question

  Let the sides of a triangle be a, b, c and the lengths of the corresponding medians be `m_a , m_b, m_c`. Show that

  `sum_{cyc} m_a / a >= {3( m_a + m_b + m_c)} /{a + b + c}`

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  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 1

• Triangle Side Lengths

  Let `n > 2` be an integer, and let ` t_1,t_2,...,t_n` be positive real numbers such that

  `(t_1+t_2+...+t_n)(1/t_1 + 1/t_2 + ... + 1/t_n) < n^2+1`

  Prove that for all i,j,k such that `1<=i<j<k<=n`, the triple of numbers `t_i,t_j,t_k` are the side lengths of a triangle.

  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Grossman Math Olympiad, 2006 Question 5

• Paths in a Triangular Park

  In a park, there are 3 straight paths that form a triangle (there are no additional paths). The entrances to the park are at the midpoints of the paths, and a lamp hangs at each vertex of the triangle. From each entrance, the shortest walking distance along the park's paths to the lamp at the opposite vertex was measured. It turned out that 2 out of the 3 distances are equal to each other. Is the triangle necessarily isosceles?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 3