Problem 6. (Option 1).

In the right triangle $ABC: \angle ACB=90^{\circ}, AC=6, BC=4$. On the line $BC$, a point $D (CD>BD)$ is marked such that $\angle ADC=45^{\circ}$. On the line $AD$, a point $E$ is marked such that the perimeter of triangle $CBE$ is the smallest possible. Then, on the line $DC$, a point $F$ is marked such that the perimeter of triangle $AFE$ is the smallest possible. Find $CF$.

![](https://cdn.mathpix.com/cropped/2024_05_06_3fdcc7ef9dfdd3dab8ecg-07.jpg?height=846&width=557&top_left_y=228&top_left_x=241)

## Construction and Proof:

For convenience, construct a square ACDK (AD is its diagonal, since $\angle ADC=45^{\circ}$. Mark point $B_1$ on side $DK (B_1D=BD \Rightarrow B_1B \perp AD)$.

Draw line $B_1C$, which intersects $AD$ at point $E$.

The perimeter of triangle $CBE$ is the smallest because among all possible points $E_1$ on line $AD$, the sum of the lengths of segments $B_1E + EC$ is the smallest ($B_1E + EC < B_1E_1 + E_1C$ - triangle inequality) and $B_1E = EB$. Similarly, mark point $A_1$ on side $AC$ ($A_1C = AC$). Draw line $A_1E$, which intersects $CD$ at point $F$.

The perimeter of triangle $AFE$ is the smallest because among all possible points $F_1$ on line $AD$, the sum of the lengths of segments $A_1F + EF$ is the smallest ($A_1F + EF < A_1F_1 + F_1E$ - triangle inequality) and $A_1F = FA$.
# Solution:

$C D=A C=6 \Rightarrow B D=B_{1} D=6-4=2$. Since triangles $\mathrm{ACE}$ and $\mathrm{B}_{1} \mathrm{ED}$

are similar ( $\angle A E C=\angle B_{1} E D$ - vertical angles and $\angle E A C=\angle B_{1} D E=45^{\circ}$ ), then

$E C: B_{1} E=A C: D B_{1}=A E: E D=6: 2=3: 1$

Let $E H \perp D C \Rightarrow$ right triangles ACD and EHD are similar ( $\angle A C D-$ common), then $C D: H D=A C: E H=A D: E D=4: 1 \Rightarrow E H=H D=1.5 \Rightarrow C H=4.5$.

Right triangles FHE and $\mathrm{FCA}_{1}$ are similar $\left(\angle E F H=\angle A_{1} F C\right.$ vertical angles) $)$

then $C F: H F=A_{1} F: E F=A_{1} C: E H=6: 1.5=4: 1 ; C H=4.5 \Rightarrow C F=0.8 \cdot 4.5=3.6$

Answer: 3.6 .