Bidyut Guha Thakurta and Amitava Mitra Solutions for Exercise 1: EXERCISE 5

In $△ABC$, $\angle A$ is right angle and in $△DBC, \angle D$ is right angle. They lie on the same side of $BC$. If $AC$ and $BD$ meet at $P$, then prove that $AP.PC=BP.PD$.

$ABCD$ is a cyclic quadrilateral. The chords $AC$ and $BD$ meet at $P$. Prove that $AP.PC=BP.PD$.

Similar Question:

If two chords of a circle intersect each other, prove that the rectangle contained by one is equal to the rectangle contained by other.

$ABCD$ is a cyclic quadrilateral. $AB$ and $DC$ produced to meet at $P$ outside the circle. Prove that $PB:PD=PC:PA$.

In a parallelogram $ABCD$ the straight line from $D$ meets $AB$ and $CB$ produced at $E$ and $F$. Prove that $AD:AE=CF:CD$.

$AB$ is a diameter of a circle. The straight line through $A$ meets the circle at $D$ and the tangent at $B$ at $C$. Prove that $B{C}^2=AC.CD$

$ABCD$ is a cyclic quadrilateral. The diagonal $BD$ bisects the diagonal $AC$. Prove that $AB.AD=CB.CD$.

ln right- angled triangle $ABC, AD$ is the perpendicular from $A$ upon the hypotenuse $BC$. Prove that $A{C}^2=BC\times CD$.

ln right- angled triangle $ABC, AD$ is the perpendicular from $A$ upon the hypotenuse $BC$. Prove that $A{D}^2=BD\times CD$.