Prove that \((a + b - c)/2 < m_c < (a + b)/2\), where \(a\), \(b\) and \(c\) are the lengths of the sides of an arbitrary triangle and \(m_c\) is the median to side \(c\).
Prove that in any triangle the sum of the lengths of the heights is less than the perimeter.
Prove that \(\angle ABC < \angle BAC\) if and only if \(AC < BC\), that is, the larger side lies opposite the larger angle of the triangle, and opposite the larger side lies the larger angle.
\(ABC\) is a right angled triangle with a right angle at \(C\). Prove that \(c^n > a^n + b^n\) for \(n > 2\).
There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
Prove that the segment connecting the vertex of an isosceles triangle to a point lying on the base is no greater than the lateral side of the triangle.