Let's modify the right-angled triangle, given that the hypotenuse and the line bisecting the right angle divides the hypotenuse in the ratio $4: 1$. Calculate the legs of the triangle!
I. Solution. The vertex $C$ of the right-angled triangle corresponding to the hypotenuse $AB$ lies on the circle constructed above the diameter $AB$. The angle bisector passes through point $E$ of this circle, which divides the semicircle on the opposite side of the diameter $AB$ into two equal parts, and it divides $AB$ at point $D$ in the ratio $4:1$.

![](https://cdn.mathpix.com/cropped/2024_05_02_e6ae3ffec56c36583666g-1.jpg?height=428&width=425&top_left_y=304&top_left_x=839)

According to the construction, it will be as follows: we divide $AB$ into two parts in the ratio $4:1$ at point $D$. We connect the midpoint of the $180^{\circ}$ arc $\widehat{AB}$ with $D$; the connecting line intersects the circle at the desired point $C$. $\triangle ABC$ meets the conditions!

According to the Angle Bisector Theorem,

$$
AC: BC = 4:1 \quad \text{that is} \quad AC = 4BC
$$

However,

$$
\begin{gathered}
\overline{AC}^2 + \overline{BC}^2 = \overline{AB}^2, \quad \text{so} \quad 16 \overline{BC}^2 + \overline{BC}^2 = \overline{AB}^2 \\
17 BC^2 = \overline{AB}^2, \quad BC = \frac{\overline{AB}}{\sqrt{17}} \quad \text{and} \quad AC = \frac{4 \overline{AB}}{\sqrt{17}}
\end{gathered}
$$

Hajdu Amadé (Bencés g. VI. o. Pápa.)

II. Solution. As we saw in the previous solution, the ratio of the legs in the desired right-angled triangle is $4:1$. All right-angled triangles in which the legs are in the ratio $4:1$ are similar.

![](https://cdn.mathpix.com/cropped/2024_05_02_e6ae3ffec56c36583666g-1.jpg?height=438&width=167&top_left_y=1445&top_left_x=968)

Therefore, let's construct a right-angled triangle with legs in the ratio $4:1$, that is,

$$
B' C' : A C' = 4:1
$$

We measure the given hypotenuse $AB$ on the hypotenuse $AB'$ and draw a perpendicular from $B$ to $AC'$; if the foot of this perpendicular is $C$, then $\triangle ABC$ is the desired right-angled triangle.