Prove that the bisectors of a triangle intersect at one point.
A circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.
Prove that the following inequalities hold for the Brockard angle \(\varphi\):
a) \(\varphi ^{3} \le (\alpha - \varphi) (\beta - \varphi) (\gamma - \varphi)\) ;
b) \(8 \varphi^{3} \le \alpha \beta \gamma\) (the Jiff inequality).
Prove that a convex quadrilateral \(ABCD\) can be drawn inside a circle if and only if \(\angle ABC + \angle CDA = 180^{\circ}\).
Prove that a convex quadrilateral \(ICEF\) can contain a circle if and only if \(IC+EH = CE+IF\).
a) Prove that the axes of symmetry of a regular polygon intersect at one point.
b) Prove that the regular \(2n\)-gon has a centre of symmetry.
a) The convex \(n\)-gon is cut by diagonals that do not cross to form triangles. Prove that the number of these triangles is equal to \(n - 2\).
b) Prove that the sum of the angles at the vertices of a convex \(n\)-gon is \((n - 2) \times 180^{\circ}\).
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 \(A\).