Determining the Coordinates of the Third Vertex of a Triangle

In geometric problems, it is often required to determine the coordinates of the third vertex of a triangle when given specific coordinates of the other two vertices and the lengths of all its sides. This phenomenon is a common issue in coordinate geometry and finds applications in various fields such as engineering, computer graphics, and surveying. This article delves into the method of finding the coordinates of the third vertex.

Given the Length of All Three Sides and Coordinates of Two Vertices

To find the coordinates of the third vertex ( (x_3, y_3) ) of a triangle, knowing the coordinates of the other two vertices ( (x_1, y_1) ) and ( (x_2, y_2) ), and the lengths of all three sides, follow these detailed steps:

Step 1: Coordinate System Initial Setup

To simplify the calculations, we can initially choose a coordinate system where:

Point 1 (( x_1, y_1 )) is the origin: ( (0, 0) ) Point 2 (( x_2, y_2 )) lies on the x-axis: ( (c, 0) )

This setup simplifies the problem by reducing the equations we need to solve.

Step 2: Equations of the Circles

The third vertex of the triangle lies at the intersection of two circles. The equations of these circles are:

Circle 1 with center at ( (0, 0) ) and radius ( a ): x^2 y^2 a^2 Circle 2 with center at ( (c, 0) ) and radius ( b ): (x - c)^2 y^2 b^2

Step 3: Solving the Equations

To find the intersection points of these circles, we solve the system of equations:

Equation 1: ( x^2 y^2 a^2 ) Equation 2: ( (x - c)^2 y^2 b^2 )

First, expand and simplify Equation 2:

Expanded Equation 2: ( x^2 - 2cx c^2 y^2 b^2 )

Subtract Equation 1 from Expanded Equation 2:

Subtraction: ( (x^2 - 2cx c^2 y^2) - (x^2 y^2) b^2 - a^2 ) Result: ( -2cx c^2 b^2 - a^2 )

Solve for ( x ):

Simplified ( x ) Equation: ( x frac{a^2 b^2 - c^2}{2c} )

Substitute ( x ) back into Equation 1 to solve for ( y ):

Substitute: ( left(frac{a^2 b^2 - c^2}{2c}right)^2 y^2 a^2 ) Expand and Simplify: ( frac{(a^2 b^2 - c^2)^2}{4c^2} y^2 a^2 ) Solve for ( y ):

The solutions for ( y ) are:

Simplified ( y ) Equations: ( y pm sqrt{a^2 - left(frac{a^2 b^2 - c^2}{2c}right)^2} )

Step 4: Applying the Triangle Inequality

To validate the solutions, ensure that they satisfy the triangle inequality:

Triangle Inequality: ( a b > c ), ( a c > b ), and ( b c > a )

Additionally, check that the imaginary part of ( y ) is zero to confirm the coordinates are real.

General Solution and Coordinate Transformation

The general solution for the coordinates of the third vertex is:

x-coordinate: ( x_3 frac{a^2 b^2 - c^2}{2c} ) y-coordinate: ( y_3 pm sqrt{a^2 - left(frac{a^2 b^2 - c^2}{2c}right)^2} )

After finding these coordinates, transform them back to the original coordinate system by applying the transformation:

Translate by the origin: ( (x_3, y_3) rightarrow (x_3 x_1, y_3 y_1) ) Apply a rotation matrix if the original coordinate system is not aligned with the axes.

Conclusion

Determining the coordinates of the third vertex of a triangle when given the lengths of its sides and the coordinates of two vertices involves solving a system of quadratic equations. The detailed method outlined in this article can help you accurately find the coordinates, ensuring that the solutions satisfy the triangle inequality and the constraints of real coordinates.