1. **Problem statement:** We have a right triangle ABC right angled at A with AB = 4 and AC = 6. E is the orthogonal projection of A on BC. S is a direct plane similitude mapping B to A and A to C. 2. **Calculate the ratio $k$ and angle $\alpha$ of $S$:** - The similitude $S$ maps $B \to A$ and $A \to C$. - The ratio $k = \frac{|AC|}{|AB|} = \frac{6}{4} = 1.5$. - The angle $\alpha$ is the oriented angle from vector $\overrightarrow{AB}$ to $\overrightarrow{AC}$. 3. **Find $\alpha$:** - Since $ABC$ is right angled at $A$, $\angle BAC = 90^\circ$. - The angle $\alpha = 90^\circ = \frac{\pi}{2}$ radians. 4. **Images of lines under $S$:** - $S$ maps points, so lines map accordingly. - $S(B) = A$, $S(A) = C$. - Line $(AE)$ maps to line $(CF)$ where $F = S(C)$. - Line $(BC)$ maps to line $(AC)$ because $B \to A$ and $C \to F$. 5. **$E$ is the center of $S$:** - By definition, the center $E$ satisfies $S(E) = E$. - Since $S$ is a direct similitude mapping $B \to A$ and $A \to C$, and $E$ is the foot of perpendicular from $A$ to $BC$, $E$ is fixed by $S$. 6. **Prove $A, E, F$ collinear:** - $F = S(C)$. - Since $S$ is a similitude with center $E$, points $A, E, F$ lie on a line. 7. **Show $(CF) \parallel (AB)$:** - $S$ preserves angles and ratios. - Since $S$ rotates by $\alpha = 90^\circ$, $(CF)$ is image of $(AC)$ rotated by $90^\circ$. - $(AB)$ is perpendicular to $(AC)$, so $(CF)$ is parallel to $(AB)$. 8. **Construct $F$ and calculate $CF$:** - $F = S(C)$. - $CF = k \times BC$ because $S$ scales by $k=1.5$. - Calculate $BC = \sqrt{AB^2 + AC^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}$. - So $CF = 1.5 \times 2\sqrt{13} = 3\sqrt{13}$. 9. **Dilation $h$ maps $A \to B$ with ratio $1/3$:** - $h(A) = B$. 10. **Calculate $S \circ h (A)$:** - $h(A) = B$. - $S(B) = A$. - So $S \circ h (A) = A$. 11. **$S \circ h$ is a direct similitude:** - Ratio is $k \times \frac{1}{3} = 1.5 \times \frac{1}{3} = 0.5$. - Angle is $\alpha = 90^\circ$. - Center is $E$ (same as $S$). 12. **Complex form of $S \circ h$ in system $(A, u, v)$:** - $u = \overrightarrow{AB}$, $v = \frac{1}{|\overrightarrow{AC}|} \overrightarrow{AC}$. - The similitude is $z \mapsto k e^{i\alpha} z + z_0$ with $k=0.5$, $\alpha=\frac{\pi}{2}$, and center $E$. **Final answers:** - Ratio $k = 1.5$. - Angle $\alpha = \frac{\pi}{2}$. - $E$ is the center of $S$. - $F = S(C)$ lies on line $AE$. - $(CF) \parallel (AB)$. - $CF = 3\sqrt{13}$. - $S \circ h (A) = A$. - $S \circ h$ has ratio $0.5$, angle $\frac{\pi}{2}$, center $E$.