Montrer-que-a-b-2-c-d-1-

Question Number 183633 by a.lgnaoui last updated on 27/Dec/22

M o n t r e r q u e

{(\frac{a}{b})}^{2} = \frac{c}{d} + 1

Commented by a.lgnaoui last updated on 27/Dec/22

Answered by mr W last updated on 28/Dec/22

Commented by mr W last updated on 28/Dec/22

R = \frac{c + d}{2}

{(R - r)}^{2} - {(R - r - d)}^{2} = b^{2}

\Rightarrow d (2 R - 2 r - d) = b^{2}

\Rightarrow d (c - 2 r) = b^{2}

\Rightarrow 2 r = c - \frac{b^{2}}{d}

{(\sqrt{a^{2} + r^{2}})}^{2} - {(R + R - r - d)}^{2} = b^{2}

a^{2} + r^{2} - {(c - r)}^{2} = b^{2}

a^{2} - c^{2} + 2 c r = b^{2}

a^{2} - c^{2} + c (c - \frac{b^{2}}{d}) = b^{2}

a^{2} = (1 + \frac{c}{d}) b^{2}

\Rightarrow {(\frac{a}{b})}^{2} = \frac{c}{d} + 1 ✓

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