[u]level C[/u]


p10. Construct the following figure, where $ABCD$ is a rectangle, $PQC$ is equilateral and $2 PD = PA$.
[img]https://cdn.artofproblemsolving.com/attachments/3/d/6fd25309c373e91b8d837df7378725b6b6fd7d.gif[/img]


p11. Let $ABC$ be an isosceles triangle, with base $AB = 10$ cm. Let $M$ and $N$ be the midpoints of the sides $AC$ and $BC$ respectively. Let $G$ be the point of intersection of $BM$ and $NA$. If the angle $AGB$ is right, find the area of ​​$ABC$.


p12. Let $A, B, C$, and $D$ be four collinear points such that $AB = BC = CD$.
Let $P$ be a point on the plane such that $2 \angle APB = \angle BPC = 2 \angle PD$.
Find the measure of angle $\angle APB$.


p13. Let S and $R$ be two circles with centers $O_1$ and $O_2$ respectively. Let $P$ be a point on $S$. A parallel to $O_1O_2$ is drawn by $P$, which intersects $R$ at $A$ and $B$. Lines $PO_1$ and $AO_2$ intersect at point $E$. Find the locus of $E$ as  $P$ moves on $S$.