第7章 7.13

(1)

$\frac{ab}{a^2+b^2} = \frac{cd}{c^2+d^2}$

$a:b=c:d$ より $\dfrac{a}{b}=\dfrac{c}{d}$

$\text{左辺} = \frac{ab}{a^2+b^2}\times\frac{\frac{1}{b^2}}{\frac{1}{b^2}} = \frac{\frac{a}{b}}{\frac{a^2}{b^2}+1} = \frac{\frac{c}{d}}{\frac{c^2}{d^2}+1} = \frac{\frac{c}{d}}{\frac{c^2+d^2}{d^2}} = \frac{cd}{c^2+d^2} = \text{右辺}$

(2)

$(a^2+b^2)(c^2+d^2) = (ac+bd)^2$

$\text{左辺} = (a^2+b^2)(c^2+d^2) = a^2c^2+a^2d^2+b^2d^2+b^2c^2 = a^2c^2+(ad\cdot ad+bc\cdot bc)+b^2d^2$

$= a^2c^2+(ad\cdot bc+bc\cdot ad)+b^2d^2 = a^2c^2+2abcd+b^2d^2 = (ac+bd)^2 = \text{右辺}$