In the right-angled triangle $ABC$ ($\angle A = 90^{\circ}$); from point $F$, we draw a circle passing through point $A$, which touches $BC$ at $B$; from point $G$, we similarly draw a circle passing through point $A$, which touches $BC$ at $C$. Tasks:

$1^{\circ}$. Show that the two circles touch each other.

$2^{\circ}$. If the sides of the right-angled triangle $ABC$ are given, calculate the radii of the circles, as well as the chords $AD$ and $AE$, which are the extensions of the sides $AC$ and $AB$.

$3^{\circ}$. If only the side $BC = a$ is given, determine the sides $AB$ and $AC$ such that $CD + BE = m$.
$1^{\circ} . D B$ and $E C$ are parallel diameters, because both are perpendicular to $B C$; therefore, $\angle F B A = \angle A E C$; but $\triangle F A B$ and $\triangle A G E$ are isosceles triangles, so $\angle F B A = \angle F A B$ and $\angle A E C = \angle G A E$; hence, $\angle F A B = \angle E A G$. $F A$ and $A G$ lie on the same straight line, which is why the circles touch each other.

![](https://cdn.mathpix.com/cropped/2024_05_02_9a0e52fe821898314a81g-1.jpg?height=398&width=542&top_left_y=305&top_left_x=781)

$2^{\circ}$. From the similarity of triangles $D B C$ and $B A C$, it follows that

$$
B D : B C = A B : A C
$$

which gives

$$
2 r_{1} = \frac{a c}{b}
$$

or

$$
r_{1} = \frac{a c}{2 b}
$$

Similarly, we find that

$$
r_{2} = \frac{a b}{2 c}
$$

From triangle $B D C$:

$$
\overline{A B}^{2} = A D \cdot A C
$$

which gives

$$
A D = \frac{c^{2}}{b}
$$

Similarly,

$$
A E = \frac{b^{2}}{c}
$$

$3^{\circ}$. Let $A B = x$ and $A C = y$.

$$
C D + B E = y + \frac{x^{2}}{y} + x + \frac{y^{2}}{x} = m
$$

which gives

$$
x \left(x^{2} + y^{2}\right) + y \left(x^{2} + y^{2}\right) = m x y
$$

or

$$
a^{2} (x + y) = m x y
$$

Since furthermore

$$
x^{2} + y^{2} = a^{2}
$$

Squaring the first equation, we get:

$$
a^{4} \left(a^{2} + 2 x y\right) = m^{2} x^{2} y^{2}
$$

which gives

$$
m^{2} x^{2} y^{2} - 2 a^{4} x y - a^{6} = 0
$$

From this equation, we can calculate $x y$, and from (1) we can find $(x + y)$, which allows us to determine $x$ and $y$.

The problem was solved by: Andráschek F., Freibauer E., Groffits G., Krisztián Gy., Lukhaub Gy., Oltay K., Perl Gy., Prohászka J., Weisz Á., Weisz J.