The triangle \(ABC\) is given. Find the locus of the point \(X\) satisfying the inequalities \(AX \leq CX \leq BX\).
Find the locus of the points \(X\) such that the tangents drawn from \(X\) to a given circle have a given length.
The point \(A\) is fixed on a circle. Find the locus of the point \(X\) which divides the chords that end at point \(A\) in a \(1:2\) ratio, starting from the point \(A\).
Let \(O\) be the center of the rectangle \(ABCD\). Find the geometric points of \(M\) for which \(AM \geq OM, BM \geq OM\), \(CM \geq OM\), and \(DM \geq OM\).
Construct a right-angled triangle along the leg and the hypotenuse.
Construct a circle with a given centre, tangent to a given circle.
Construct a straight line passing through a given point and tangent to a given circle.
Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).
Construct the triangle ABC by the medians \(m_a, m_b\) and \(m_c\).
Solve the equations \(x^2 = 14 + y^2\) in integers.