Geometric Proof of Heron's Formula Using Coordinate Geometry

Introduction

Heron's formula is a classical method to calculate the area of a triangle when the lengths of all three sides are known. While the traditional proof relies on trigonometry, this article explores an alternative approach by utilizing coordinate geometry. This method provides a unique perspective on the relationship between the vertices of a triangle and its area.

Step 1: Placing the Triangle in the Coordinate Plane

Consider a triangle with vertices at the coordinates:

A(0, 0) B(a, 0) C(x, y)

Here, the length of side AB is a.

Step 2: Calculating the Lengths of the Other Sides

To find the lengths of sides AC and BC, we use the distance formula:

AC b sqrt{x^2 y^2} BC c sqrt{(x - a)^2 y^2}

These lengths are derived from the coordinates of the points involved.

Step 3: Using the Determinant Formula to Calculate the Area

The area A of triangle ABC can be calculated using the determinant formula based on the coordinates of its vertices:

A frac{1}{2} |x_1(y_2 - y_3) x_2(y_3 - y_1) x_3(y_1 - y_2)|

Substituting the coordinates of points A(0, 0), B(a, 0), and C(x, y) into the formula:

A frac{1}{2} |0 cdot (0 - y) a cdot (y - 0) x cdot (0 - 0)| A frac{1}{2} |ay|

Step 4: Relating the Area to the Sides Using Heron's Formula

To express the area in terms of a, b, and c, we use the Law of Cosines to derive the relationship between these sides and y:

b^2 x^2 y^2 c^2 (x - a)^2 y^2 x^2 - 2ax a^2 y^2

By equating the two expressions for y^2:

b^2 - x^2 c^2 - x^2 - 2ax a^2

Simplifying the equation:

2ax c^2 - b^2 a^2

From this, we can solve for x and then find the value of y^2 in terms of a, b, and c:

y^2 c^2 - x^2 - a^2 2ax

Step 5: Final Steps

After some algebraic manipulation, we can express the area in a form that aligns with Heron's formula:

A sqrt{ss - as - bs - c}

wheres frac{a b c}{2}.

Conclusion

By utilizing coordinate geometry and the relationships derived from the triangle's vertices, we have demonstrated that the area calculated using the determinant method aligns with Heron's formula, thus providing a geometric proof of the formula.