# vector

AP

= sAB + (1-s)AN = sAB + (1-s)AC * 3/5 ・・・(1)

= tAM + (1-t)AC  = tAB * 2/3 + (1-t)AC・・・(2)

where s and t are parameters.

comparing coefficients of (1) and (2) expressions,

(AB and AC are linearly independence)

s = t*2/3 and

(1-s)*3/5= 1-t

solving these equations,

s = 4/9

t = 2/3

substituting s = 4/9 into the expression (1),

AP = 4/9*AB + 1/3*AC

AQ

= uAB + (1-u)AC ・・・(a)

= vAP

= 4v/9*AB +  v/3*AC・・・(b)

comparing coefficietns of (a) and (b),

u = 4v/9 and

1-u = v/3

solving these equations,

u = 4/7,

v = 9/7

substituting u = 4/7 into the expression (a),

AQ = 4/7*AB + 3/7*AC

# formularization

let AB and AC be AB !// AC ∧ AB!=0 ∧ AC!=0.

AP

= sAB + (1-s)AN = sAB + (1-s)AC * β

= tAM + (1-t)AC  = tAB * α + (1-t)AC

comparing coefficients of these two expressions,

s   = tα and

(1-s)β  = 1-t

solving these equations,

s = (α-αβ)/(1-αβ)

t = (1-β)/(1-αβ)

AP =

(α-αβ)         (β-αβ)

------- AB + ------- AC

(1-αβ)         (1-αβ)

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