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Question Number 203018 by mr W last updated on 07/Jan/24

Commented by mr W last updated on 07/Jan/24

$$\left[anunsolvedoldquestionQ202864\right]$$

Commented by esmaeil last updated on 07/Jan/24

a

Commented by MathematicalUser2357 last updated on 09/Jan/24

$$anunquolved202864$$

Answered by mr W last updated on 07/Jan/24

Commented by mr W last updated on 07/Jan/24

$$wecanseethatthethreegreenlines$$$$areparalleltoeachother.$$$$let\alpha =\frac{q}{p},\beta =\frac{r}{p}$$$$1+\frac{2825+B}{690+A}={\left(\frac{q+r}{p}\right)}^2={(\alpha +\beta )}^2$$$$\Rightarrow A=\frac{2825+B}{{(\alpha +\beta )}^2-1}-690$$$$1+\frac{B}{46}={\left(\frac{p}{q}\right)}^2=\frac{1}{{\alpha }^2}\Rightarrow \alpha =\frac{1}{\sqrt{1+\frac{B}{46}}}$$$$\frac{B}{2825}=\frac{p+q}{r}=\frac{1+\alpha }{\beta }\Rightarrow \beta =\frac{2825}{B}(1+\frac{1}{\sqrt{1+\frac{B}{46}}})$$$$1+\frac{2825\times 2+B}{A}={\left(\frac{q+2r}{p}\right)}^2={(\alpha +2\beta )}^2$$$$\Rightarrow A=\frac{2825\times 2+B}{{(\alpha +2\beta )}^2-1}$$$$\frac{2825+B}{{(\alpha +\beta )}^2-1}-690=\frac{2825\times 2+B}{{(\alpha +2\beta )}^2-1}$$$$\Rightarrow B\approx 1593.2072$$$$\Rightarrow A\approx 412.5313$$$$?=A+B\approx 2005.7385✓$$

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