Problems

Ten straight lines are drawn through a point on a plane cutting the plane into angles.
Prove that at least one of these angles is less than ${20}^{\circ }$.

One of the four angles formed when two straight lines intersect is ${41}^{\circ }$. What are the other three angles equal to?

In an isosceles triangle, the sides are equal to either 3 or 7. Which side length is the base?

In the acute-angled triangle $ABC$, the heights $A{A}_1$ and $B{B}_1$ are drawn. Prove that $A_{1}C\times BC=B_{1}C\times AC$.

Let $A{A}_1$ and $B{B}_1$ be the heights of the triangle $ABC$. Prove that the triangles $A_1{B}_{1}C$ 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 $\mathrm{\angle }BAH=\mathrm{\angle }OAC$.

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 $\mathrm{\angle }BAH=\mathrm{\angle }OAC$.

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 $\mathrm{\angle }PAK=\mathrm{\angle }MAQ$.

On a circle, the points $A,B,C,D$ are given in the indicated order. $M$ is the midpoint of the arc $AB$. We denote the intersection points of the chords $MC$ and $MD$ with the chord $AB$ by $E$ and $K$. Prove that $KECD$ is an inscribed quadrilateral.