§ 3

Find the area of a triangle if its sides lie on the lines with the equations $x+y=1, x-y=-1$ and $2 x+y=2$ in a Cartesian rectangular coordinate system.

Suppose that under an affine transformation the points $A, B$ and $C$ are carried into the points $A^{*}, B^{*}$ and $C^{*}$. Prove that the point of intersection of the medians of $\triangle A B C$ is carried into the point of intersection of the medians of $\triangle A^{*} B^{*} C^{*}$.

We shall say that an affine transformation stretches a vector $\mathbf{a}$ by a factor of $\alpha$ if $\left|\mathbf{a}^{*}\right|=\alpha \left|\mathbf{a}\right|$. For the transformation given in a Cartesian rectangular coordinate system by the formulas

find the vectors for which the stretching is

Suppose a line is tangent to a second-order curve. Prove that under an arbitrary affine transformation the image of the line is tangent to the image of the curve.

Prove that the vertices of a rhombus circumscribed about an ellipse lie on its axes of symmetry.

Represent as a product of two axial symmetries

Represent the compression toward the axis of abscissas of a Cartesian rectangular coordinate system as a product of the translation by $\mathbf{a}(0, a)$ and a compression toward another line.

Can the product of two rotations of the plane be a translation?