Step 1: Write the coordinates
Let
P be any point
(x,y).A is
(1,0).B is
(0,−2).C is
(2,−1).
Step 2: Find area of triangle
PAB The area of triangle
PAB is given by:
Area of
△PAB=‌|x1(y2−y3)+x2(y3−y1)+x3(y1−y2)| For
P(x,y),A(1,0),B(0,−2) :
‌ Area ‌=‌|x(0+2)+1(−2−y)+0(y−0)|=‌|2x−2−y| Step 3: Find area of triangle
PAC Use the same formula for
P(x,y),A(1,0),C(2,−1) :
‌ Area ‌=‌|x(0+1)+1(−1−y)+2(y−0)|=‌|x+y−1| Step 4: Set the areas equal
We want these two areas to be the same:
‌|2x−2−y|=‌|x+y−1| Take both sides times
2:|2x−2−y|=|x+y−1| Step 5: Remove the modulus (absolute value)
Square both sides to get rid of absolute values:
(2x−2−y)2=(x+y−1)2 Step 6: Expand both sides
Expand the left side:
(2x−2−y)2=(2x−y−2)2=4x2+y2+4+4xy−8x−4y Expand the right side:
(x+y−1)2=x2+y2+1+2xy−2x−2y Step 7: Move all terms to one side
Subtract right side from left side:
‌(4x2+y2+4+4xy−8x−4y)−(x2+y2+1+2xy−2x−2y)=0‌ Simplify: ‌ ‌4x2−x2+4xy−2xy−8x+2x+(−4y+2y)+4−1=03x2+2xy−6x−2y+3=0 Step 8: Divide by 3
To make it simpler, divide all terms by 3:
x2−2xy−2x+2y+1=0