Kerala Board Solutions for Chapter: Circles, Exercise 3: Exercise 3

Calculate the angles of the quadrilateral in the picture given below, and also the angles between their diagonals,

 

Prove that any outer angle of a cyclic quadrilateral is equal to the inner angle at the opposite vertex.

Prove that a parallelogram which is not a rectangle is not cyclic.

Prove that any non-isosceles trapezium is not cyclic.

In the picture given below, bisectors of adjacent angles of the quadrilateral $\mathrm{ABCD}$ intersect at $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$. Prove that $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$ is a cyclic quadrilateral.

The two circles below intersect at $P, Q$ and lines through these points meet the circles at $A, B, C, D.$ The lines $AC$ and $BD$ are not parallel. Prove that if these lines are of equal length, then $ABDC$ is a cyclic quadrilateral.

In the picture, the circles on the left and right intersect the middle circle at $P, Q, R, S;$ the lines joining them meet the left and right circles at $A, B, C, D.$ Prove that $ABCD$ is a cyclic quadrilateral.

In the picture, points $P, Q, R$ are marked on the sides $BC, CA, AB$ of $∆ABC$ and the circumcircles of $∆AQR$ and $∆BRP$ are drawn. $\mathrm{M}$ is a point where these circles intersect. Prove that the circumcircle of $∆CPQ$ also passes through $\mathrm{M}.$