Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.
One of the four angles formed when two straight lines intersect is \(41^{\circ}\). What are the other three angles equal to?
Let \(AA_1\) and \(BB_1\) be the heights of the triangle \(ABC\). Prove that the triangles \(A_1B_1C\) and \(ABC\) are similar.
The bisector of the outer corner at the vertex \(C\) of the triangle \(ABC\) intersects the circumscribed circle at the point \(D\). Prove that \(AD = BD\).
The vertex \(A\) of the acute-angled triangle \(ABC\) is connected by a segment with the center \(O\) of the circumscribed circle. The height \(AH\) is drawn from the vertex \(A\). Prove that \(\angle BAH = \angle OAC\).
The vertex \(A\) of the acute-angled triangle \(ABC\) is connected by a segment with the center \(O\) of the circumscribed circle. The height \(AH\) is drawn from the vertex \(A\). Prove that \(\angle BAH = \angle OAC\).
From an arbitrary point \(M\) lying within a given angle with vertex \(A\), the perpendiculars \(MP\) and \(MQ\) are dropped to the sides of the angle. From point \(A\), the perpendicular \(AK\) is dropped to the segment \(PQ\). Prove that \(\angle PAK = \angle MAQ\).
Prove that from the point \(C\) lying outside of the circle we can draw exactly two tangents to the circle and the lengths of these tangents (that is, the distance from \(C\) to the points of tangency) are equal.
Two circles intersect at points \(A\) and \(B\). Point \(X\) lies on the line \(AB\), but not on the segment \(AB\). Prove that the lengths of all of the tangents drawn from \(X\) to the circles are equal.
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 \((a + b + c)/2\).