$AB$ is a diameter of a circle. $CD$ is a chord parallel to $AB$ and $2CD=AB$. The tangent at $B$ meets the line $AC$ produced at $E$. Prove that $AE=2AB$.

If the points $\left({\alpha }_1,{\beta }_1\right),\left({\alpha }_2,{\beta }_2\right),\left({\alpha }_3,{\beta }_3\right)$ lies on the circle $x^2+y^2+2gx+2fy=0$, then prove that the four straight lines
${\alpha }_{1}x+{\beta }_{1}y-m\left({\alpha }_1^2+{\beta }_1^2\right)=0$

${\alpha }_{2}x+{\beta }_{2}y-m\left({\alpha }_2^2+{\beta }_2^2\right)=0$

${\alpha }_{3}x+{\beta }_{3}y-m\left({\alpha }_3^2+{\beta }_3^2\right)=0,fx-gy=0$
meet in a point.

Prove that the equation of the circumcircle of the triangle formed by the lines

$u_1\equiv a_{1}x+b_{1}y+c_1=0$

$u_2\equiv a_{2}x+b_{2}y+c_2=0$

$u_3\equiv a_{3}x+b_{3}y+c_3=0$

is $\left|\begin{array}{ccc}\frac{1}{u_1}& \frac{1}{u_2}& \frac{1}{u_3}\\ a_2{a}_3-b_2{b}_3& a_3{a}_1-b_3{b}_1& a_1{a}_2-b_1{b}_2\\ a_2{b}_3+a_3{b}_2& a_3{b}_1+a_1{b}_3& a_1{b}_2+a_2{b}_1\end{array}\right|=0$

or $\left|\begin{array}{ccc}\frac{a_1^2+b_1^2}{u_1}& \frac{a_2^2+b_2^2}{u_2}& \frac{a_3^2+b_3^2}{u_3}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|=0$

Prove that the general equation of circles cutting the circles $x^2+y^2+2g_{i}x+2f_{I}y+c_i=0;i=1,2$ orthogonally is

$\left|\begin{array}{lll}{x}^2+y^2& -x& -y\\ c_1& g_1& f_1\\ c_2& g_2& f_2\end{array}\right|+k\left|\begin{array}{ccc}-x& -y& 1\\ g_1& f_1& 1\\ g_2& f_2& 1\end{array}\right|=0$

An isosceles right-angled triangle, whose sides are $1,1,\sqrt{2}$ lies entirely in the first quadrant with the ends of the hypotenuse on the co-ordinate axes. If it slides, prove that the locus of its centroid is

${\left(3x-y\right)}^2+{\left(x-3y\right)}^2=\frac{32}{9}$