# 三角形计算器

## 解决方案

1. $A = \left(\frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right)^0$

第三个角是$B = 180^0 - \left(A + C\right)$

在我们的例子中， $B = 180^0 - \left(\left(\frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right)^0 + 45^0\right) = \left(\frac{- \pi \left(45 + \frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right) + 180 \pi}{\pi}\right)^0$

面积为$S = \frac{1}{2} a b \sin{\left(C \right)} = \left(\frac{1}{2}\right)\cdot \left(9\right)\cdot \left(10\right)\cdot \left(\sin{\left(45^0 \right)}\right) = \frac{45 \sqrt{2}}{2}$

周长是$P = a + b + c = 9 + 10 + \sqrt{181 - 90 \sqrt{2}} = \sqrt{181 - 90 \sqrt{2}} + 19$

2. $A = \left(\frac{- 180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)} + 180 \pi}{\pi}\right)^0$

第三个角是$B = 180^0 - \left(A + C\right)$

在我们的例子中， $B = 180^0 - \left(\left(\frac{- 180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)} + 180 \pi}{\pi}\right)^0 + 45^0\right) = \left(\frac{- \pi \left(45 + \frac{- 180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)} + 180 \pi}{\pi}\right) + 180 \pi}{\pi}\right)^0$

这种情况是不可能的，因为与较长边相对的角度必须更大。

## 回答

$a = 9$A

$b = 10$A

$c = \sqrt{181 - 90 \sqrt{2}}\approx 7.329446049083208$A

$A = \left(\frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right)^0\approx 60.258581489369345^0$A

$B = \left(\frac{- \pi \left(45 + \frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right) + 180 \pi}{\pi}\right)^0\approx 74.741418510630655^0$A

$C = 45^0$A