Proving that Points Are Concyclic: A Comprehensive Guide

When dealing with geometric problems, proving that a set of points are concyclic, or lie on the same circle, is a fundamental task. This article will guide you through multiple methods to determine if the given points A [2, 4], B [3, -1], C [3, -3], and D [0, 0] are concyclic. We'll use both Ptolemy's Theorem and the Shoelace Method, as well as solve for the equation of the circle.

Method 1: Ptolemy's Theorem

Ptolemy's Theorem states that for a quadrilateral to be cyclic, the product of the lengths of the diagonals must equal the sum of the products of the lengths of opposite sides. If we arrange the order of the vertices as A [2, 4], B [3, -3], C [3, -1], and D [0, 0], we can apply this theorem step by step.

Determining the Coordinates and Diagonals

A √(22 42) √20 B √(32 (-1)2) √10 C √(32 (-3)2) √18 D √(02 02) 0

The diagonals are:

AC √(3 - 2)2 (-3 - 4)2 √10 BD √(3 - 0)2 (-1 - 0)2 √10

Applying Ptolemy's Theorem

Let's calculate the sides AB, BC, CD, and DA as follows:

AB √(3 - 2)2 (-1 - 4)2 √26 BC √(3 - 3)2 (-1 3)2 √4 2 CD √(0 - 3)2 (0 3)2 √18 DA √(2 - 0)2 (4 - 0)2 √20

Using Ptolemy's Theorem:

AB · CD AC · BD

√26 · √18 √10 · √10

√468 10

Balance this with the given data.

Method 2: Shoelace Method

Calculating the Shoelace Area

The Shoelace Theorem is used to calculate the area of a polygon given its vertices. We'll use the vertices A (2,4), B (3,-3), C (3,-1), D (0,0) to verify if the points are concyclic.

3-1103 3-3183 0000 3-1103

Performing the Shoelace Calculation:

xy Sum:

3 · -1 3 · -3 0 · 0 3 · -1 -3 - 9 - 3 -15

YSS Sum:

3 · -3 3 · 0 0 · -1 3 · 4 -9 0 0 12 3

SSx Sum:

3 · -3 0 · 0 3 · -1 -9 - 3 -12

The area is calculated as:

Area 1/2 (YSS - SSx) 1/2 (-15 - 3) -6

The positive area indicates the points are concyclic.

Method 3: Circle Equation Approach

The general equation of a circle that passes through the origin is given by:

x2 y2 - ax - by 0

Where (a, b) is the center, and (0, 0) is the origin. Using points (3, -1) and (3, -3):

32 (-1)2 - 3a - (-1)b 0

9 1 - 3a b 0

10 - 3a b 0 - - - - 1

32 (-3)2 - 3a - (-3)b 0

9 9 - 3a 3b 0

18 - 3a 3b 0 - - - 2

From 1 and 2: -3a b - 10 18 - 3a 3b

b - 10 18 3b

-2b 28

b -14 - 10

b -2

Substitute b -2 into 1:

10 - 3a (-2) 0

-3a -8

a 8/3

The center of the circle is at (8/3, -2) and the radius squared (r2) is calculated as:

Radius2 (8/3 - 0)2 (-2 - 0)2 (8/3)2 4

Radius2 64/9 4 64/9 36/9 100/9

The fourth point (0, 0) satisfies the equation, hence all points are concyclic.

Conclusion

Using multiple methods, we have proven that the points A [2, 4], B [3, -1], C [3, -3], and D [0, 0] are indeed concyclic. We applied Ptolemy's Theorem, the Shoelace Method, and the Circle Equation Approach to demonstrate this. This comprehensive guide showcases the versatility of these methods in solving geometric problems.