For a triangle $ABC$, the following are known:

$$
a=7.012, \quad c-b=1.753 \quad \text{and} \quad B=38^{\circ} 12^{\prime} 48^{\prime \prime}
$$

Calculate the sides and angles of the triangle. 1 ![^0]


[^0]:    1*) The formulas should be reduced to a form suitable for logarithmic calculation.
Let $a+b-c=2 p$, then $a-b+c=2(p-b)$ and $a+b-c=2(p-c)$. Furthermore, the radius of the inscribed circle of a triangle $r=(p-b) \tan \frac{B}{2}=(p-c) \tan \frac{C}{2}=(p-a) \tan \frac{A}{2}$, and the necessary formulas for calculation are as follows:

$$
\begin{gathered}
\log r=\log (p-b)+\log \tan \frac{B}{2} \\
\log \tan \frac{C}{2}=\log r-\log (p-c) \\
\frac{A}{2}=90^{\circ}-\frac{B+C}{2} \\
\log (p-a)=\log r+\log \cot \frac{A}{2}
\end{gathered}
$$

# Auxiliary calculations:

$$
\begin{aligned}
&(p-b)=4,3825 \\
& \log (p-b)=0,641722 \\
&(p-c)=2,6925 \\
& \log (p-c)=0,419873 \\
& \hline
\end{aligned}
$$

Calculation of $\log \tan \frac{B}{2}$:

| $\frac{B}{2}=$ | $19^{\circ} 6^{\prime} 23,75^{\prime \prime}$ |
| ---: | ---: |
| $\lg \left(19^{\circ} 0^{\prime} 20^{\prime \prime}\right)$ | $9,539565-10$ |
| $3{ }^{\prime \prime}$ | 20 |
| $0,7^{\prime \prime}$ | 4,8 |
| $0,0^{\prime \prime}$ | 0,34 |

$\overline{\log \tan \frac{B}{2}=9,539590-10}$

Calculation of $\log \cot \frac{A}{2}$:

$$
\lg \tan 40^{\circ} 53^{\prime} 30^{\prime \prime} \quad 9,937504-10
$$

$$
\begin{array}{rr}
6^{\prime \prime} & 25 \\
0.25^{\prime \prime} & 1 \\
\hline \log \tan \frac{A}{2}=9,937530-10 \\
\log \cot \frac{A}{2} \quad=0,062470
\end{array}
$$

## Final calculations:

Calculation of $\log r$:

0,641722

9,53959

$\overline{\log r=0,181312}$

Calculation of $C$ and $A$:

| $-\quad 0,181312$ |
| ---: | ---: |
| $-\quad 0,419873$ |
| $\log \tan \frac{C}{2}=9,761439-10$ |

$$
\begin{gathered}
\frac{C}{2}=30^{\circ}, \frac{A}{2}=40^{\circ} 53^{\prime} 36,25^{\prime \prime} \\
C=60^{\circ}, A=81^{\circ} 47^{\prime} 12,5^{\prime \prime}
\end{gathered}
$$

Calculation of $b$ and $c$:

$$
\begin{array}{rr} 
& 0,181312 \\
0,062470 \\
\hline \log (p-a)= & 0,243782 \\
p-a & =1,7530 \\
p & =8,7650 \\
b & =4,3825 \\
c & =6,1355
\end{array}
$$

## Results:

$$
\begin{gathered}
A=81^{\circ} 47^{\prime} 12,45^{\prime \prime} \\
C=60^{\circ} \\
b=4,3825 \\
c=6,1355
\end{gathered}
$$

The problem was solved by: Friedmann Bernát, S.-A.-Ujhely; Grünhut Béla, Pécs; Ifj. Imre János, Nyíregyháza; Jankovich György, Losoncz; Jorga Gergely, Gilád; Krausz Mihály, Budapest; Meitner Elemér, Budapest; Stramkó Loránd, Rimaszombat; Főgymnasiumi VII. osztálya S.-A.-Ujhely; Weisz Lipót and Visnya Aladár, Pécs.