Given-in-an-isosceles-triangle-a-lateral-side-b-and-the-base-angle-Compute-the-distance-from-the-centre-of-the-inscribed-circle-to-the-centre-of-the-circumscribed-circle-

Question Number 19586 by ajfour last updated on 13/Aug/17

$$\mathrm{Given}\:\mathrm{in}\:\mathrm{an}\:\mathrm{isosceles}\:\mathrm{triangle}\:\mathrm{a} \\ $$$$\mathrm{lateral}\:\mathrm{side}\:\mathrm{b}\:\mathrm{and}\:\mathrm{the}\:\mathrm{base}\:\mathrm{angle}\:\alpha. \\ $$$$\mathrm{Compute}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{inscribed}\:\mathrm{circle}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circumscribed}\:\mathrm{circle}. \\ $$

Commented by Tinkutara last updated on 13/Aug/17

$${R}\:=\:\frac{{b}}{\mathrm{2}\:\mathrm{sin}\:\alpha}\:,\:{r}\:=\:\frac{{b}\:\mathrm{cos}\:\alpha}{\mathrm{cos}\:\frac{\alpha}{\mathrm{2}}} \\ $$

Commented by ajfour last updated on 13/Aug/17

Commented by Tinkutara last updated on 13/Aug/17

$$\mathrm{I}\:\mathrm{found}\:\mathrm{this}\:\mathrm{on}\:\mathrm{Wikipedia}\:\mathrm{by}\:\mathrm{Euler}'\mathrm{s} \\ $$$$\mathrm{Theorem}\:\mathrm{in}\:\mathrm{Geometry}. \\ $$$$\mid{IO}\mid\:=\:\sqrt{{R}\left({R}\:−\:\mathrm{2}{r}\right)}.\:\mathrm{Proof}\:\mathrm{is}\:\mathrm{given}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{page}. \\ $$

Commented by ajfour last updated on 13/Aug/17

$$\mathrm{required}\:\mathrm{to}\:\mathrm{find}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\mathrm{b},\:\mathrm{and}\:\alpha\:. \\ $$

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