Let $A = (-1,-1)$, $B = (3,2)$ and $C = (11,4)$ be points of the euclidean plane.
(a) Give the equattion of the line $C$ in Hesse normal form and calculate the diastance between $B$ and the line $AC$.
(b) Give the reflection $\sigma$ to the line $AC$ in analytic form.
(c) Let $\tau$ be the reflection at the line $AB$ and $\rho$ the reflection at the line $BC$. What linear trnaformation is $\sigma\circ\tau$ and $\sigma\circ\tau$ and $\sigma\circ\tau\circ\rho$ (translation, rotation or similar) ? Justify your answer.
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I have done (a) and (b).
For (c) I have done the following:
The lines $AC$ and $AB$ intersect in $A$.
The composition of $\sigma$ and $\tau$ is then a rotation around the intersection point, i.e. $A$, with angle $2\theta$, where $\theta$ is the angle between $AB$ and$AC$.
The composition of $\sigma$, $\tau$ and $\rho$, i.e. the composition of a rotation and a reflection is a reflection.
Is that correct? How can we justify that further?