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### Finding the Coordinates of a Triangle Vertex

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

Respected sir,

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
```

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

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/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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