The bisector of the outer corner at the vertex \(C\) of the triangle \(ABC\) intersects the circumscribed circle at the point \(D\). Prove that \(AD = BD\).
Let \(a\) and \(b\) be the lengths of the sides of a right-angled triangle and \(c\) the length of its hypotenuse. Prove that:
a) The radius of the inscribed circle of the triangle is \((a + b - c)/2\);
b) The radius of the circle that is tangent to the hypotenuse and the extensions of the sides of the triangle, is equal to \((a + b + c)/2\).
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 \(\angle ABC + \angle CDA = 180^{\circ}\).
The triangle \(ABC\) is given. Find the locus of the point \(X\) satisfying the inequalities \(AX \leq CX \leq BX\).
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) \(\sqrt {ab}\).
Prove that \(\angle ABC > 90^{\circ}\) if and only if the point \(B\) lies inside a circle with diameter \(AC\).
Two circles of radius \(R\) intersect at points \(D\) and \(B\). Let \(F\) and \(G\) be the points of intersection of the middle perpendicular to the segment \(BD\) with these circles lying on one side of the line \(BD\). Prove that \(BD^2 + FG^2 = 4R^2\).
A parallelogram \(ABCD\) and a point \(E\) are given. Through the points \(A, B, C, D\), lines parallel to the straight lines \(EC, ED, EA,EB\), respectively, are drawn. Prove that they intersect at one point.