Problems

A circle is divided up by the points $A,B,C,D$ so that $⌣AB:⌣BC:⌣CD:⌣DA=2:3:5:6$. The chords $AC$ and $BD$ intersect at point $M$. Find the angle $AMB$.

A circle is divided up by the points $A$, $B$, $C$, $D$ so that $⌣AB:⌣BC:⌣CD:⌣DA=3:2:13:7$. The chords $AD$ and $BC$ are continued until their intersection at point $M$. Find the angle $AMB$.

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.

Two circles intersect at the points $P$ and $Q$. Through the point $A$ of the first circle, the lines $AP$ and $AQ$ are drawn intersecting the second circle at points $B$ and $C$. Prove that the tangent at point $A$ to the first circle is parallel to the line $BC$.

On the circle, the points $A,B,C$ and $D$ are given. The lines $AB$ and $CD$ intersect at the point $M$. Prove that $AC\times AD/AM=BC\times BD/BM$.

In the triangle $ABC$, the height $AH$ is drawn; $O$ is the center of the circumscribed circle. Prove that $\mathrm{\angle }OAH=|\mathrm{\angle }B-\mathrm{\angle }C$|.