Problems

Problem 22   A circle $k$ is tangent to a line $l$ at a point $P$. Let $O$ be diametrically opposite to $P$ on $k$. For some points $T,S\in k$ set $OT\cap l={T_1}$ and $OS\cap l={S_1}$. Finally, let $SQ$ and $TQ$ be two tangents to $k$ meeting in point $Q$. Set $OQ\cap l=\{Q_1\}$. Prove that $Q_1$ is the midpoint of $T_1S_1$.

Problem 23   Consider $\triangle ABC$ and its circumscribed and inscribed circles $K$ and $k$, respectively. Take an arbitrary point $A_1$ on $K$, draw through $A_1$ a tangent line to $k$ and let it intersect $K$ in point $B_1$. Now draw through $B_1$ another tangent line to $k$ and let it intersect $K$ in point $C_1$. Finally, draw through $C_1$ a third tangent line to $k$ and let it intersect $K$ in point $D_1$ (cf. Fig. 9.) Prove that $D_1$ coincides with $A_1$. In other words, prove that any triangle $A_1B_1C_1$ inscribed in $K$, two of whose sides are tangent to $k$, must have its third side also tangent to $k$ so that $k$ is the inscribed circle for $\triangle A_1B_1C_1$ too.

Problem 24   Find the distance between the center $P$ of the inscribed circle and the center $O$ of the circumscribed circle of $\triangle ABC$ in terms of the two radii $r$ and $R$.

Problem 25   We are given $\triangle ABC$ and points $D\in AC$ and $E\in BC$ such that $DE\vert\vert AB$. A circle $k_1$ of diameter $DB$ intersects a circle $k_2$ of diameter $AE$ in $M$ and $N$. Prove that $M$ and $N$ lie on the altitude $CH$ to $AB$.

Problem 26   Prove that the altitude of $\triangle ABC$ through $C$ is the radical axis of the circles with diameters the medians $AM$ and $BN$ of $\triangle ABC$.

Problem 27   Find the geometric place of points $O$ which are centers of circles through the end points of diameters of two fixed circles $k_1$ and $k_2$.

Problem 28   Construct all radical axes of the four incircles of $\triangle ABC$.

Problem 29   Let $A,B,C$ be three collinear points with $B$ inside $AC$. On one side of $AC$ we draw three semicircles $k_{1},k_{2}$ and $k_{3}$ with diameters $AC$, $AB$ and $BC$, respectively. Let $BE$ be the interior tangent between $k_{2}$ and $k_{3}$ ( $E\in k_{1}$), and let $UV$ be the exterior tangent to $k_{2}$ and $k_{3}$ ( $U\in k_{2}$ and $V\in k_{3}$). Find the ratio of the areas of $\triangle UVE$ and $\triangle ACE$ in terms of $k_{2}$ and $k_{3}$'s radii.(cf. Fig. 10)

Problem 30   The chord $AB$ separates a circle $\gamma$ into two parts. Circle $\gamma_{1}$ of radius $r_{1}$ is inscribed in one of the parts and it touches $AB$ at its midpoint $C$. Circle $\gamma_{2}$ of radius $r_{2}$ is also inscribed in the same part of $\gamma$ so that it touches $AB$, $\gamma_{1}$ and $\gamma$. Let $PQ$ be the interior tangent of $\gamma_{1}$ and $\gamma_{2}$, with $P,Q\in\gamma$. Show that $PQ\cdot SE=SP\cdot SQ$, where $S=\gamma_{1}\cap \gamma_{2}$ and $E=AB\cap PQ$.(cf. Fig. 11)

Problem 31   Let $k_{1}(O,R)$ be the circumscribed circle around $\triangle ABC$, and let $k_{2}(T,r)$ be the inscribed circle in $\triangle ABC$. Let $k_{3}(T,r_1)$ be a circle such that there exists a quadrilateral $AB_{1}C_{1}D_{1}$ inscribed in $k_{1}$ and circumscribed around $k_{3}$. Calculate $r_{1}$ in terms of $R$ and $r$.

Problem 32   Let $ABCD$ be a square, and let $l$ be a line such that the reflection $A_{1}$ of $A$ across $l$ lie on the segment $BC$. Let $D_{1}$ be the reflection of $D$ across $l$, and let $D_{1}A_{1}$ intersect $DC$ in point $P$. Finally, let $k_{1}$ be the circle of radius $r_{1}$ inscribed in $\triangle A_{1}CP_{1}$. Prove that $r_{1}=D_{1}P_{1}$.

Problem 33   In a circle $k(O,R)$ let $AB$ be a chord, and let $k_{1}$ be a circle touching internally $k$ at point $K$ so that $KO\perp AB$. Let a circle $k_{2}$ move in the region defined by $AB$ and not containing $k_{1}$ so that it touches both $AB$ and $k$. Prove that the tangent distance between $k_{1}$ and $k_{2}$ is constant.

Problem 34   Prove that for any two circles there exists an inversion which transforms them into congruent circles (of the same radii). Prove further that for any three circles there exists an inversion which transforms them into circles with collinear centers.

Problem 35   Given two nonintersecting circles $k_{1}$ and $k_{2}$, show that all circles orthogonal to both of them pass through two fixed points and are tangent pairwise.

Problem 36   Given two circles $k_{1}$ and $k_{2}$ intersecting at points $A$ and $B$, show that there exist exactly two points in the plane through which there passes no circle orthogonal to $k_{1}$ and $k_{2}$.