Problems

Find the locus of the midpoints of the segments, the ends of which are found on two given parallel lines.

The triangle $ABC$ is given. Find the locus of the point $X$ satisfying the inequalities $AX\le CX\le BX$.

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 $A$.

Construct the triangle $ABC$ along the side $a$, the height $h_a$ and the angle $A$.

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) $\sqrt{ab}$.

A wide variety of questions in mathematics starts with the question ’Is it possible...?’. In such problems you would either present an example, in case the described situation is possible, or rigorously prove that the situation is impossible, with the help of counterexample or by any other means. Sometimes the border between what seems should be possible and impossible is not immediately obvious, therefore you have to be cautious and verify that your example (or counterexample) satisfies the conditions stated in the problem. When you are asked the question whether something is possible or not and you suspect it is actually possible, it is always useful to ask more questions to gather additional information to narrow the possible answers. You can ask for example "How is it possible"? Or "$\mathbf{W}\mathbf{h}\mathbf{i}\mathbf{c}\mathbf{h}$ properties should the correct construction satisfy"?