Finding the Coordinates of a Triangle Vertex

Date: 10/26/1999 at 12:49:37
From: madhusudhan
Subject: Analytical geometry problem

Respected sir,

I am unable to solve this problem. Please help me out. The problem is:

The points B and C of triangle ABC lie on the line 3y = 4x and y = 0 
respectively. The line BC passes through (2/3,2/3). If O is the origin 
and AOBC forms a rhombus, find the co-ordinates of A.

I tried to solve this problem by drawing a diagram, but I could not 
get any ideas. Please help me out.

Date: 10/26/1999 at 14:36:03
From: Doctor Rob
Subject: Re: Analytical geometry problem

Thanks for writing to Ask Dr. Math, Madhusudhan.

The coordinates of B must be (3*a,4*a) for some a, because B lies on 
3*y = 4*x. The coordinates of C must be (b,0) for some b, because C 
lies on y = 0.  The line BC has equation

   (y-0)*(3*a-b) = (x-b)*(4*a-0)

       (3*a-b)*y = 4*a*(x-b)

Since the point (2/3,2/3) lies on it, we get an equation relating
a and b:

   (3*a-b)*(2/3) = 4*a*(2/3-b)

Next, since AOBC is a rhombus, the lengths of its sides must be equal. 
You know the coordinates of B, C, and O(0,0), and those of A are found 
from the parallelogram rule. Then you can compute the lengths of all 
the sides, and setting these expressions equal will give an additional 
equation relating a and b.

Solve the two equations relating a and b together simultaneously, and 
you'll get the values of a and b. Substitute these into the 
expressions for the coordinates of A, and you'll be done.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/