In Fig. 6.62, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that

(i) Δ PAC ~ Δ PDB

(ii) PA . PB = PC . PD

In Fig. 6.63, D is a point on side BC of Δ ABC such that $\dfrac{BD}{CD} = \dfrac{AB}{AC}$ Prove that AD is the bisector of $\angle BAC$.

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (see Fig. 6.64)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?

In Fig. 6.56, PS is the bisector of $\angle QPR$ of Δ PQR. Prove that $\dfrac{QS}{SR} = \dfrac{PQ}{PR}$

In Fig. 6.57, D is a point on hypotenuse AC of Δ ABC, such that BD $\perp$ AC, DM $\perp$ BC and DN $\perp$ AB. Prove that :

(i) $DM^2 = DN . MC $

(ii) $DN^2 = DM . AN $

In Fig. 6.58, ABC is a triangle in which $\angle ABC > 90°$ and AD $\perp$ CB produced. Prove that $AC^2 = AB^2 + BC^2 + 2 BC . BD.$

In Fig. 6.59, ABC is a triangle in which $\angle ABC < 90°$ and AD $\perp$ BC. Prove that $AC^2 = AB^2 + BC^2 – 2 BC.BD.$

In Fig. 6.60, AD is a median of a triangle ABC and AM ⊥ BC. Prove that :

(i)$ AC^2 = AD^2 + BC . DM + (\dfrac{BC}{2})^2$

(ii) $AB^2 = AD^2 – BC . DM + (\dfrac{BC}{2})^2$

(iii) $AC^2 + AB^2 = 2 AD^2 + \dfrac{1}{2} BC^2$

Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

In Fig. 6.61, two chords AB and CD intersect each other at the point P. Prove that :

(i) Δ APC ~ Δ DPB

(ii) AP . PB = CP . DP