<span><span><span><span><span><span>обозначим середину ребра МА точкой K.</span></span></span></span></span></span>
О - точка пересечения диагоналей основания, т.е. МО - высота пирамиды;
точка E - середина MB, точка G - середина MD;
P - точка пересечения диагоналей KGCE;
KH - перпендикуляр, опущенный от точки K на прямую AC
<span><span><span><span><span><span>M</span><span>O</span></span></span></span></span></span> и <span><span><span><span><span><span>C</span><span>K -</span></span></span></span></span></span> медианы треуголника <span><span><span><span><span><span>A</span><span>M</span><span>C</span><span>⇒</span><span><span><span><span><span>M</span><span>P/</span></span></span><span><span><span>M</span><span>O</span></span></span></span></span><span>=</span><span><span><span><span>2/</span></span><span><span>3</span></span></span></span></span></span></span></span></span>
<span><span><span><span><span><span>△</span><span>M</span><span>E</span><span>G</span><span>∼<span>△</span></span><span>M</span><span>B</span><span>D</span><span>⇒</span><span>E</span><span>G</span><span>=</span><span><span><span><span>2/</span></span><span><span>3</span></span></span></span><span>B</span><span>D</span><span>.</span></span></span></span></span></span>
<span><span><span><span><span><span><span><span><span><span><span><span>E</span><span>K</span><span>G</span><span>C</span></span></span></span></span></span><span> - четырехугольник, диагонали которого - перпендикулярны: </span><span><span><span><span><span><span>B</span><span>D</span><span>⊥</span><span>M</span><span>O</span><span>,</span><span>B</span><span>D</span><span>⊥</span><span>A</span><span>C</span><span>⇒</span><span>B</span><span>D</span><span>⊥</span><span>(</span><span>M</span><span>A</span><span>C</span><span>)</span><span>⇒</span><span>B</span><span>D</span><span>⊥</span><span>C</span><span>K</span><span>.</span></span></span></span></span></span></span></span></span></span></span></span>
Так как <span><span><span><span><span><span>E</span><span>G</span><span><span><span>|</span></span></span><span><span><span>|</span></span></span><span>B</span><span>D</span><span>, значит </span><span>E</span><span>G</span><span>⊥</span><span>C</span><span>K</span></span></span></span></span></span>
<span><span><span><span><span><span>C</span><span>H</span><span>=</span><span><span><span><span>3/</span></span><span><span>4</span></span></span></span><span>A</span><span>C</span><span>,</span><span>K</span><span>H</span><span>=</span><span><span><span><span>1/</span></span><span><span>2</span></span></span></span><span>M</span><span>O ( KH - средняя линия треугольника AMC)</span><span>, </span><span>C</span><span>K</span><span>= (</span><span><span><span><span><span>C</span><span><span><span><span>H^</span></span><span><span>2</span></span></span></span><span>+</span><span>K</span><span><span><span><span>H^</span></span><span><span>2) под корнем ( по теореме Пифагора) </span></span></span></span></span></span></span></span></span></span></span></span></span>
<span><span><span><span><span><span><span><span><span><span><span><span><span>( а дальше я не знаю...) </span></span></span></span></span></span></span></span></span></span></span></span></span>