Find the ratio of the legs of a right-angled triangle if it is known that the touching point

Find the ratio of the legs of a right-angled triangle if it is known that the touching point of the inscribed circle divides the hypotenuse in a ratio of 2: 3.

Let us denote the right-angled triangle given by the condition ABC, angle C – straight line, AB – hypotenuse.
The circle is inscribed in a triangle, its center is the intersection point of the bisectors.
The distance from the vertex of the corner to the point of tangency of the circle and the side of the triangle are equal.
Let’s write down the sides of the triangle:
Hypotenuse AB = 2 + 3 = 5.
BC leg = 2 + x.
AC leg = 3 + x.
We write down the Pythagorean theorem:
AB² = AC² + BC²
25 = (3 + x) ² + (2 + x) ²
2x² + 10x – 12 = 0
x² + 5x – 6 = 0
x1 = -6, x2 = 1.
-6 is an extraneous root.
BC = 2 + 1 = 3;
AC = 3 + 1 = 4.
We find the ratio of the legs:
BC / AC = 3/4.
Answer: 3/4.