Example 36. In triangle $A B C \widehat{C}=90^{\circ},[A D)$ and $[B K)-$ are the bisectors of the interior angles. Find $\widehat{B A C}$ and $\widehat{A B C}$, if $2|A D| \cdot|B K|=|A B|^{2}$
The solution is (Fig. 40). Introduce an auxiliary parameter, setting $|A B|=c$, and let $\widehat{D A C}=x$. Then $\widehat{B A C}=2 x, \widehat{A B C}=$ $=90^{\circ}-2 x$ and $\widehat{C B} K=45^{\circ}-x$.

From the right triangle $A B C$, we find that $|A C|=$ $=c \cos 2 x$. From the right triangle $A D C$, we have:

$$
|A D|=\frac{|A C|}{\cos x} .
$$

Substituting the found value of $|A C|$, we get:

$$
|A D|=\frac{c \cos 2 x}{\cos x}
$$

Similarly, from the right triangle $A B C$, we find that $|B C|=c \sin 2 x$, and from the right triangle $B C K$, we get:

$$
|B K|=\frac{|B C|}{\cos \left(45^{\circ}-x\right)}=\frac{c \sin 2 x}{\cos \left(45^{\circ}-x\right)}
$$

According to the problem's condition, we write the following equation:

$$
\frac{2 c \cos 2 x}{\cos x} \cdot \frac{c \sin 2 x}{\cos \left(45^{\circ}-x\right)}=c^{2}
$$

or

$$
\frac{2 \sin 2 x \cos 2 x}{\cos x \cos \left(45^{\circ}-x\right)}=1
$$

Since $\cos 2 x=\sin \left(90^{\circ}-2 x\right)=2 \sin \left(45^{\circ}-x\right) \cos \left(45^{\circ}-x\right)$, and $\sin 2 x=2 \sin x \cos x$, we get:

$$
8 \sin \left(45^{\circ}-x\right) \sin x=1
$$

from which

$$
4\left(\cos \left(45^{\circ}-2 x\right)-\cos 45^{\circ}\right)=1
$$

Therefore,

$$
\cos \left(2 x-45^{\circ}\right)=\frac{1+2 \sqrt{2}}{4}
$$

Thus,

$$
2 x=45^{\circ}+\arccos \frac{1+2 \sqrt{2}}{4}
$$

Investigating this. By the problem's meaning, $0^{\circ}<2 x<90^{\circ}$. The found value of $2 x$ satisfies this system of inequalities, since

$$
\frac{\sqrt{2}}{2}<\frac{1+2 \sqrt{2}}{4}<1
$$

i.e.,

$$
0^{\circ}<\arccos \frac{1+2 \sqrt{2}}{4}<45^{\circ}
$$

Thus, we have:

$$
\widehat{B A C}=45^{\circ}+\arccos \frac{1+2 \sqrt{2}}{4}, \widehat{A B C}=45^{\circ}-\arccos \frac{1+2 \sqrt{2}}{4}
$$

## Problems for Independent Solution

1. The median to the hypotenuse of a right triangle, which divides the angle in the ratio $1: 2$, is equal to $m$. Find the sides of the triangle.
2. The perimeter of a parallelogram is 44 cm, and it is divided by the diagonals into four triangles such that the difference between the perimeters of adjacent triangles is 6 cm. Determine the sides of the parallelogram.
3. The diagonal of a rectangular trapezoid is congruent to its lateral side and is equal to 4 cm. Find the length of the midline of the trapezoid if its height is 2 cm.
4. The larger base of a trapezoid is $24 \mathrm{~cm}$, and the distance between the midpoints of its diagonals is 4 cm. Find the smaller base of the trapezoid.
5. The height of a rhombus divides its side into segments of lengths $m$ and $n$. Find the diagonals of the rhombus.
6. The lateral side of an isosceles triangle is 4 cm, and the median to the lateral side is 3 cm. Find the base.
7. The base of an isosceles triangle is $4 \sqrt{2}$, and the median to the lateral side is 5 cm. Find the lateral side.
8. In a rectangular trapezoid, the bases and the smaller lateral side are equal to $a, b$ and $c$, respectively. Find the distances from the point of intersection of the diagonals to the bases and the smaller lateral side.
9. A square is inscribed in a right triangle with legs equal to $a$ and $b$, and it shares a right angle with the triangle. Find the perimeter of the square.

10(1). A rhombus is inscribed in a right triangle such that all its vertices lie on the sides of the triangle, and the angle equal to $60^{\circ}$ is a common angle of the rhombus and the triangle. Find the sides of the triangle if the side of the rhombus is 6 cm.

11. A rhombus is inscribed in a triangle such that one angle is common to both, and the opposite vertex of the rhombus divides the side of the triangle into segments whose lengths are in the ratio $2: 3$. Find the sides of the triangle between which the common angle of the triangle and the rhombus lies, if the diagonals of the rhombus are $m$ and $n$.

12(2). One of the angles of a trapezoid is $30^{\circ}$, and the lateral sides, when extended, intersect at a right angle. Find the smaller lateral side of the trapezoid if its midline and one of the bases are equal to 10 and 8 cm, respectively.

13. In triangle $A B C|A C|=b,|A B|=c$, and the angle bisector of $\angle B A C$ intersects side $B C$ at point $D$ such that $|A D|=$ $=|B D|$. Find side $B C$.

14(3). From the vertex of the right angle of a triangle, a bisector is drawn, dividing the hypotenuse into segments of lengths $m$ and $n$. Find the height dropped from the vertex of the right angle.

15(4). In triangle $A B C|B C|=a,|A C|=b, \angle