[u]level 2[/u]


[b]p5. [/b]Given a circle $\omega$, a point $A$ in its interior and a point $B$ on $\omega$, find points $C$ and $D$ in $\omega$ such that $ABCD$ is a parallelogram.


[b]p6.[/b] Given a parallelogram $ABCD$, let $A'$ in $AB$ such that$ BA' = 3AA'$, let $B'$ in $BC$ such that $CB'= 3BB'$, let $C'$ in $CD$ such that $DC' = 3CC'$, let $D' $ in $DA$ such that $AD'= 3DD'$. Let $A''$ in $D'A'$ such that $D'A''= 3A'A'$, let $B''$ in $A'B'$ such that $A'B'' = 3B'B''$, let $C''$  in $B'C'$ such that $B'C''= 3C'C''$, let $D''$ in $C'D'$ such that $C'D''= 3D'D''$. Prove that $A''B''C''D''$ is similar to $ABCD$ and that they are parallel. Find their ratio of similarity.


[b]p7. [/b]In a triangle, draw the segments that join each vertex with the incenter. The three bisectors of these segments determine a triangle. Prove that it is inscribed in the circle circumscribed to the original triangle.


[b]p8. [/b]Given a circle $\omega$ and a point $P$ inside it, construct the triangles $APB$ with $A$ and $B$ in $\omega$ and $\angle APB=90^o$.
$\bullet$ Find the locus of the midpoint $M$ of segment $AB$.
$\bullet$ Find the locus of the foot $H$ of the altitude from $P$.
1. **Construct the line $\ell$ through the center of the circle $\omega$ such that $\ell \parallel AB$.**
   - Let $O$ be the center of the circle $\omega$.
   - Draw the diameter $XY$ of $\omega$ such that $XY \parallel AB$.

2. **Reflect point $A$ over the line $\ell$ to get point $A'$.**
   - Since $A$ is inside the circle, its reflection $A'$ will also be inside the circle.
   - The reflection $A'$ is such that $A$ and $A'$ are symmetric with respect to $\ell$.

3. **Construct the circle $\omega'$ with the same radius as $\omega$ and passing through points $A$ and $A'$.**
   - The center of $\omega'$ will be the midpoint of $AA'$.

4. **Find the intersection points $C_1$ and $C_2$ of circles $\omega$ and $\omega'$.**
   - Since $\omega$ and $\omega'$ have the same radius and intersect at $A$ and $A'$, they will intersect at two other points $C_1$ and $C_2$.

5. **Determine points $D_1$ and $D_2$ such that $ABC_1D_1$ and $ABC_2D_2$ are parallelograms.**
   - For $ABC_1D_1$ to be a parallelogram, $D_1$ must be such that $AD_1 \parallel BC_1$ and $AB \parallel C_1D_1$.
   - Similarly, for $ABC_2D_2$ to be a parallelogram, $D_2$ must be such that $AD_2 \parallel BC_2$ and $AB \parallel C_2D_2$.

6. **Verify the construction:**
   - Since $C_1$ and $C_2$ are on $\omega$, they are equidistant from the center $O$.
   - The points $D_1$ and $D_2$ can be found by translating $C_1$ and $C_2$ by the vector $\overrightarrow{AB}$.

Thus, the points $C_1, D_1$ and $C_2, D_2$ are the required points such that $ABCD$ is a parallelogram.

The final answer is $\boxed{C_1, D_1 \text{ and } C_2, D_2}$