
5. There is $|B C|=1$ in a triangle $A B C$ and there is a unique point $D$ on $B C$ such that $|D A|^{2}=|D B| \cdot|D C|$. Find all possible values of the perimeter of $A B C$.

(Patrik Bak)


Solution. Let us denote by $E$ the second intersection of $A D$ with the circumcircle $k$. The power of $D$ with respect to $k$ gives $|D B| \cdot|D C|=|D A| \cdot|D E|$, which together with the given condition $|D A|^{2}=|D B| \cdot|D C|$ yields $|D A|=|D E|$. That is $E$ lie the image $p$ of the line $B C$ in the homothety with center $A$ and a coefficient 2 (Fig.1).

Vice versa, to any intersection of a line $p$ with the circle $k$ we reconstruct the point $D$ on $B C$, which fulfills $|D A|^{2}=|D B| \cdot|D C|$.

If the reconstruction have to be unique, the line $p$ has to touch $p$ in $E$.

![](https://cdn.mathpix.com/cropped/2024_04_17_d252d571aa2423c54010g-18.jpg?height=369&width=424&top_left_y=1051&top_left_x=289)

Fig. 2

![](https://cdn.mathpix.com/cropped/2024_04_17_d252d571aa2423c54010g-18.jpg?height=367&width=422&top_left_y=1050&top_left_x=790)

Fig. 3

Let us denote $S_{b}$ and $S_{c}$ in order the centers of $A C$ and $A B$. The homothety with the center $A$ and a coefficient $\frac{1}{2}$ sends $A, B, C, E$ (lying on the circle $k$ ) to $A, S_{c}$, $S_{b}, D$ which lie on the circle $k^{\prime}$ (Fig. 2), while the image of $p$ is the tangent $B C$ of $k^{\prime}$ in $D$. The powers of $B$ and $C$ with respect to $k^{\prime}$ give $|B D|^{2}=|B A| \cdot\left|B S_{c}\right|=\frac{1}{2}|B A|^{2}$ and $|C D|^{2}=|C A| \cdot\left|C S_{b}\right|=\frac{1}{2}|C A|^{2}$. All together for the perimeter of $A B C$ :

$$
|B C|+|A B|+|A C|=|B C|+\sqrt{2}(|B D|+|C D|)=|B C|+\sqrt{2} \cdot|B C|=1+\sqrt{2},
$$

which is the only possible value.
