Proving Points Concyclic Using Determinants

Understanding Concyclic Points: In geometry, points are said to be concyclic if they all lie on the same circle. This article will guide you through the process of proving whether the given points P(3, 3), Q(7, 1), R(4, 0), and S(6, 4) are concyclic using a determinant-based method. The key is to use the property that four points are concyclic if the determinant of a specific matrix is zero.

Step 1: Setting Up the Matrix

To determine if the points are concyclic, we utilize the following determinant property:

det begin{vmatrix} x_1 y_1 x_1^2 y_1^2 1 x_2 y_2 x_2^2 y_2^2 1 x_3 y_3 x_3^2 y_3^2 1 x_4 y_4 x_4^2 y_4^2 1 end{vmatrix} 0

In this case, the coordinates of the points are:

P(3, 3) gives x_1 3, y_1 3, x_1^2 9, y_1^2 9 Q(7, 1) gives x_2 7, y_2 1, x_2^2 49, y_2^2 1 R(4, 0) gives x_3 4, y_3 0, x_3^2 16, y_3^2 0 S(6, 4) gives x_4 6, y_4 4, x_4^2 36, y_4^2 16

Step 2: Construct the Determinant

Sustituting these values into the matrix:

det begin{vmatrix} 3 3 9 9 1 7 1 49 1 1 4 0 16 0 1 6 4 36 16 1 end{vmatrix}

Step 3: Calculate the Determinant

The next step is to calculate the determinant. This can be done using expansion by minors or utilizing a computational tool. For the sake of accuracy, let's use a computational method.

Calculation:

The determinant can be expanded as follows:

det 3 begin{vmatrix} 1 49 1 0 16 1 4 36 1 end{vmatrix} - 3 begin{vmatrix} 7 49 1 4 16 1 6 36 1 end{vmatrix} - 18 begin{vmatrix} 7 1 1 4 0 1 6 4 1 end{vmatrix} - 1 begin{vmatrix} 7 1 49 1 4 0 16 1 6 4 36 1 end{vmatrix}

Each of these 3x3 determinants can be computed as:

begin{vmatrix} 1 49 1 0 16 1 4 36 1 end{vmatrix} 1(16 - 16) - 49(0 - 16) 1(0 - 64) 0 784 - 64 720 begin{vmatrix} 7 49 1 4 16 1 6 36 1 end{vmatrix} 7(16 - 36) - 49(4 - 6) 1(144 - 96) -112 98 48 34 begin{vmatrix} 7 1 1 4 0 1 6 4 1 end{vmatrix} 7(0 - 4) - 1(4 - 24) 1(16 - 0) -28 20 16 8 begin{vmatrix} 7 1 49 1 4 0 16 1 6 4 36 1 end{vmatrix} 7(0 - 64) - 1(4 - 24) 49(16 - 0) - 1(32 - 64) -448 20 784 32 428

Thus, the final calculation is:

det 3(720) - 3(34) - 18(8) - 428 2160 - 102 - 144 - 428 1586

Conclusion:

As the determinant does not equal zero, P, Q, R, and S are not concyclic. This method provides a clear and straightforward way to verify concyclicity using determinants.

Additional Techniques: Midpoints and Steiner Point

Another method of determining if points are concyclic involves checking midpoints. The midpoints of PQ and RS were calculated and found to be the same, indicating a potential Steiner point. However, this alone is not sufficient to prove concyclicity; one must still use the determinant method as shown.