Sides of two similar triangles are in the ratio $4:9$. Areas of these triangles are in the ratio

In the figure,two chords $AB$ and $CD$ intersect each other at the point $P$. Prove that $AP \cdot PB = CP \cdot DP$.

In the figure,$AD$ is a median of a triangle $ABC$ and $AM \perp BC$. Prove that:
$AB^{2} = AD^{2} - BC \cdot DM + \left(\frac{BC}{2}\right)^{2}$

State which pairs of triangles in the figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.

Fill in the blanks using the correct word given in brackets:
$(i)$ All circles are $........$ (congruent,similar)
$(ii)$ All squares are $.........$ (similar,congruent)
$(iii)$ All $.........$ triangles are similar. (isosceles,equilateral)
$(iv)$ Two polygons of the same number of sides are similar,if $(a)$ their corresponding angles are $......$ and $(b)$ their corresponding sides are $......$ (equal,proportional)

In the figure,altitudes $AD$ and $CE$ of $\Delta ABC$ intersect each other at the point $P$. Show that $\Delta ABD \sim \Delta CBE$.