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Question Number 186030 by Rupesh123 last updated on 31/Jan/23

Answered by ajfour last updated on 31/Jan/23

$$x^2+y^2=d^2$$$$(a-x)y=d^2$$$$(b-y)x=d^2$$$$d(\frac{x}{y}+1+\frac{y}{x})=c$$$$\Rightarrow ay=bx$$$$\Rightarrow \frac{a}{b}+1+\frac{b}{a}=\frac{c}{d}$$

Answered by mr W last updated on 31/Jan/23

$$generallyforanytriangle:$$$$\frac{ch}{2}=\mathrm{\Delta }=\frac{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}{4}$$$$h=\frac{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}{2c}$$$$\frac{h-d}{h}=\frac{d}{c}$$$$\Rightarrow \frac{1}{d}=\frac{1}{c}+\frac{1}{h}$$$$\Rightarrow \frac{1}{d}=\frac{1}{c}+\frac{2c}{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}$$$$$$$$incaseofrightangledtriangle:$$$$h=\frac{ab}{c}$$$$\Rightarrow \frac{1}{d}=\frac{1}{c}+\frac{c}{ab}$$

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