Kaj je ortocenter trikotnika s koti (7, 8), (3, 4) in (8, 3) #?

Naj bodo koordinate treh tock trikotnika ABC

#A -> (7,8) "" B -> (3,4) "" C -> (8,3) #

Naj koordinata#color (rdeča) ("Ortho center O" -> (h, k)) #

#m_ (AB) -> "Nagib AB" = ((8-4)) / ((7-3)) = 1 #

#m_ (BC) -> "Nagib BC" = ((4-3)) / ((3-8)) = - 1/5 #

#m_ (CO) -> "Nagib CO" = ((k-3)) / ((h-8)) #

#m_ (AO) -> "Nagib AO" = ((k-8)) / ((h-7)) #

Če je ortocenter, bo premica, ki poteka skozi C in O, pravokotna na AB, Torej #m_ (CO) xxm_ (AB) = - 1 #

# => ((k-3)) / ((h-8)) xx 1 = -1

# => k = -h + 11 …. (1) #

Ko je orthocenter, bo premica, ki poteka skozi A in O, pravokotna na BC, Torej #m_ (AO) xxm_ (BC) = - 1 #

# => ((k-8)) / ((h-7)) xx (- 1/5) = - 1 #

# => k = 5h-27 …. (2) #

Primerjava (1) in (2)

# 5h-27 = -h + 11 #

# => 6h = 38 #

# => h = 6 1/3 #

Vstavljanje vrednosti h v (1)

# k = -6 1/3 + 11 = 4 2/3 #

Zato je koordinata ortocentra

#barva (zelena) ((6 1/3 "," 4 2/3)) #