Задача 29553 Зная высоты треугольника, найдите его...

slava191

2018-09-09 13:26:08

Зная высоты треугольника, найдите его площадь. [9.57]

sova

2018-09-09 14:15:33

★

S=(1/2)a*h_(a) ⇒ a=2S/h_(a)
S=(1/2)b*h_(b) ⇒ b=2S/h_(b)
S=(1/2)c*h_(c) ⇒ c=2S/h_(c)

По формуле ГЕРОНА:

S= sqrt(p*(p-a)*(p-b)*(p-c))

p=((2S/h_(a))+(2S/h_(b))+(2S/h_(c)))/2=S*((1//h_(a))+(1/h_(b))+(1/h_(c))

p-a=S*((1//h_(a))+(1/h_(b))+(1/h_(c)) - (2S/h_(a))=

= S*((1/h_(b))+(1/h_(c) -(1//h_(a)) )

p-b=S*((1//h_(a))+(1/h_(b))+(1/h_(c)) - (2S/h_(b))=

= S*((1/h_(a))+(1/h_(c) -(1//h_(b)) )

p-c=S*((1//h_(a))+(1/h_(b))+(1/h_(c)) - (2S/h_(c))=

= S*((1/h_(a))+(1/h_(b) -(1//h_(c)) )

S^2= p*(p-a)*(p-b)*(p-c)

S^2=S^4*((1//h_(a))+(1/h_(b))+(1/h_(c))*((1/h_(b))+(1/h_(c) -(1//h_(a)) ) *((1/h_(a))+(1/h_(c) -(1//h_(b)) )*((1/h_(a))+(1/h_(b) -(1//h_(c)) )

S^2=1: ((1//h_(a))+(1/h_(b))+(1/h_(c))*((1/h_(b))+(1/h_(c) -(1//h_(a)) ) *((1/h_(a))+(1/h_(c) -(1//h_(b)) )*((1/h_(a))+(1/h_(b) -(1//h_(c)) )

Значит, чтобы найти S извлекаем корень из выражения справа.