In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. Then,$BC$ is the geometric mean of..........

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AB = 10.8$,$MB = 4.5$ and $AC = 4.8$,find $AN$.

$O$ is the point of intersection of the diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$. Through $O$,a line segment $PQ$ is drawn parallel to $AB$ meeting $AD$ in $P$ and $BC$ in $Q$. Prove that $PO = QO$.

Is it true to say that if in two triangles,an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle,then the triangles are similar? Give reasons for your answer.

In $\triangle PQR$,$PD \perp QR$ such that $D$ lies on $QR$. If $PQ = a$,$PR = b$,$QD = c$,and $DR = d$,prove that $(a+b)(a-b) = (c+d)(c-d)$.

In parallelogram $ABCD$,$AB^{2} + BC^{2} = 260$ and $AC = 18$. Then $BD = \ldots$