MKH

r

O

HM

a

HK^{2}-HM^{2}=HM^{2}-MK^{2}

OLK

L

MKH

\frac{a}{2\sqrt{3}}\cdot\sqrt{r^{2}-\frac{a^{2}}{3}}

HK=b

KM=c

b^{2}-a^{2}=a^{2}-c^{2}~\Rightarrow~b^{2}+c^{2}=2a^{2}.

KK_{1}

MKH

KK_{1}^{2}=\frac{1}{4}(2b^{2}+2c^{2}-a^{2})=\frac{1}{4}(4a^{2}-a^{2})=\frac{3}{4}a^{2}.

KL=\frac{2}{3}KK_{1}=\frac{2}{3}\cdot\frac{a\sqrt{3}}{2}=\frac{a\sqrt{3}}{3}.

L

MKH

\overrightarrow{OL}=\frac{1}{3}(\overrightarrow{OM}+\overrightarrow{OK}+\overrightarrow{OH}).

\angle KOM=2\gamma,~\angle KOH=2\beta,~\angle MOH=2\alpha.

MKH

180^{\circ}

OL^{2}=\overrightarrow{OL}^{2}=\frac{1}{9}(\overrightarrow{OM}^{2}+\overrightarrow{OK}^{2}+\overrightarrow{OH}^{2}+2\overrightarrow{OM}\cdot\overrightarrow{OK}+2\overrightarrow{OK}\cdot\overrightarrow{OH}+2\overrightarrow{OM}\cdot\overrightarrow{OH})=

=\frac{1}{9}(r^{2}+r^{2}+r^{2}+2r\cdot r\cdot\cos2\gamma+2r\cdot r\cdot\cos2\beta+2r\cdot r\cdot\cos2\alpha)=

=\frac{1}{9}r^{2}(3+2(\cos2\gamma+\cos2\beta+\cos2\alpha))=

=\frac{1}{9}r^{2}(3+2(1-2\sin^{2}\gamma+1-2\sin^{2}\beta+1-2\sin^{2}\alpha))=

=\frac{1}{9}r^{2}(9-4(\sin^{2}\gamma+\sin^{2}\beta+\sin^{2}\alpha))=

=\frac{1}{9}r^{2}\left(9-4\left(\left(\frac{c}{2r}\right)^{2}+\left(\frac{b}{2r}\right)^{2}+\left(\frac{a}{2r}\right)^{2}\right)\right)=

=r^{2}-\frac{1}{9}\cdot4\cdot\frac{1}{4}(a^{2}+b^{2}+c^{2})=r^{2}-\frac{1}{9}\cdot3a^{2}=r^{2}-\frac{a^{2}}{3}.

OL=\sqrt{r^{2}-\frac{a^{2}}{3}}

KL=\frac{a\sqrt{3}}{3}

OK=r

OLK

OL^{2}+KL^{2}=\left(r^{2}-\frac{a^{2}}{3}\right)+\frac{a^{2}}{3}=r^{2}=OK^{2},

OLK

KL

OL

S_{\triangle OLK}=\frac{1}{2}\cdot KL\cdot OL=\frac{1}{2}\cdot\frac{a\sqrt{3}}{3}\cdot\sqrt{r^{2}-\frac{a^{2}}{3}}=\frac{a}{2\sqrt{3}}\cdot\sqrt{r^{2}-\frac{a^{2}}{3}}.