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Question Number 76717 by peter frank last updated on 29/Dec/19

A t r i a n g l e i s f o r m e d b y

t h e t h r e e s t r a i g h t l i n e

y = m_{1} x + \frac{a}{m_{1}}

y = m_{2} x + \frac{a}{m_{2}}

y = m_{3} x + \frac{a}{m_{3}}

p r o v e t h a t i t s o r t h o c e n t e r

a l w a y s l i e s o n t h e l i n e

x + a = 0

Answered by mr W last updated on 30/Dec/19

i n t e r s e c t i o n o f l i n e 1 a n d l i n e 2 : P

y_{P} = m_{1} x_{P} + \frac{a}{m_{1}} = m_{2} x_{P} + \frac{a}{m_{2}}

\Rightarrow x_{P} = \frac{a}{m_{1} m_{2}}

\Rightarrow y_{P} = \frac{a (m_{1} + m_{2})}{m_{1} m_{2}}

i n t e r s e c t i o n o f l i n e 1 a n d l i n e 3 : Q

\Rightarrow x_{Q} = \frac{a}{m_{1} m_{3}}

\Rightarrow y_{Q} = \frac{a (m_{1} + m_{3})}{m_{1} m_{3}}

p e r p e n d i c u l a r l i n e f r o m P t o l i n e 3 :

y = - \frac{1}{m_{3}} (x - x_{P}) + y_{P}

y = - \frac{1}{m_{3}} (x - x_{P}) + y_{P}

p e r p e n d i c u l a r l i n e f r o m Q t o l i n e 2 :

y = - \frac{1}{m_{2}} (x - x_{Q}) + y_{Q}

o r t h o c e n t e r M :

y_{M} = - \frac{1}{m_{3}} (x_{M} - x_{P}) + y_{P} = - \frac{1}{m_{2}} (x_{M} - x_{Q}) + y_{Q}

(m_{3} - m_{2}) x_{M} = m_{3} x_{Q} - m_{2} x_{P} + m_{2} m_{3} (y_{Q} - y_{P})

(m_{3} - m_{2}) x_{M} = \frac{a}{m_{1}} - \frac{a}{m_{1}} + \frac{a (m_{1} + m_{3}) m_{2}}{m_{1}} - \frac{a (m_{1} + m_{2}) m_{3}}{m_{1}}

(m_{3} - m_{2}) x_{M} = a (m_{2} - m_{3})

\Rightarrow x_{M} = - a

y_{M} = - \frac{1}{m_{3}} (- a - \frac{a}{m_{1} m_{2}}) + \frac{a (m_{1} + m_{2})}{m_{1} m_{2}}

\Rightarrow y_{M} = (\frac{1 + m_{1} m_{2} + m_{2} m_{3} + m_{3} m_{1}}{m_{1} m_{2} m_{3}}) a

i . e . i n d e p e n d e n t f r o m m_{1}, m_{2}, m_{3}

t h e x - c o o r d i n a t e o f t h e o r t h o c e n t e r

i s a l w a y s x_{M} = - a . t h a t m e a n s i t

a l w a y s l i e s o n t h e l i n e x + a = 0 .

Commented by peter frank last updated on 30/Dec/19

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