Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
Suppose you're at a park and there are three attractions you want to see. If you place the map of the park on a grid and let your starting point (the parking lot) be at the point (0,0), then your starting point and the attractions fall at the following points:
(0,0) (1,-2) (4,-4) (9,6)
It's a beautiful day, so you want to know if it's possible to walk a longer circular path that will visit all of the attractions rather than walking straight to each attraction. In other words, you want to know if there's a circle that all four points lie on.
In mathematics, we call points that lie on the same circle, and are therefore the same distance from the center of the circle, concyclic points. Therefore, mathematically speaking, we want to know if the points (0,0), (1,-2), (4,-4), and (9,6) are concyclic points. Let's explore!
Proving Concyclic Points
Proving that a set of given points are concyclic sounds harder than it really is, depending on how many points are in the set. We have some really nice theorems to help us determine if a set of 2, 3, or 4 points are concyclic.
It's pretty easy to determine that any two points are concyclic. To illustrate this, consider the fact that if we draw a line between any two points, that line can be a diameter of a circle, and both of its endpoints would lie on a circle. Therefore, any two points are concyclic.
When it comes to a set of three points, we have a nice theorem that allows us to determine (or prove) whether or not the three points are concyclic. That theorem states the following:
- Theorem: Any three points that are non-collinear (meaning they don't lie on the same line) are concyclic.
This is because if we connect any three non-collinear points with line segments, we form a triangle, and all triangles can be inscribed in a circle.
Therefore, given a set of 3 points, we can prove they are concyclic by determining that they don't all lie on the same line.
This is great so far, but what we're most interested in is our four points in the park and whether or not they are concyclic. Thankfully, a Greek writer and mathematician, Claudius Ptolemy, created a theorem that makes proving that four points are concyclic much easier than one would expect.
- Ptolemy's Theorem: A quadrilateral can be drawn in a circle if and only if the product of the measures of its diagonals are equal to the sum of the products of the measures of the pairs of opposite sides.
Since any four points determine a quadrilateral, we can use Ptolemy's Theorem to determine, or prove, that any four points are concyclic by:
- Finding the product of the lengths of the diagonals of the quadrilateral formed by the points.
- Finding the sum of the products of the measures of the pairs of opposite sides of the quadrilateral formed by the points.
- If these two values are equal, the points are concyclic. If they're not equal, then the points are not concyclic.
Great! Let's figure out if we can enjoy the day with a circular walk to visit these park attractions.
First, we draw our quadrilateral that our points form.
To perform steps one and two, we're going to need to use the distance formula, which states that the distance between the points (x1, y1) and (x2, y2) is given by the formula:
d = √((x2 - x1)2 + (y2 - y1)2)
We notice that the diagonals of the quadrilateral have endpoints (1,-2), (9,6) and (0,0), (4,-4), so we use the distance formula to find the distance between each of these pairs of points and multiply those distances together to find their product.
We get that the product of the lengths of the diagonals is 64. On to step 2!
Now we use the distance formula to find the lengths of each of the sides of the quadrilateral, multiply the lengths of opposite sides together, and find the sum of these two products.
What do you know? We get that the sum of the products of the lengths of opposite sides is also equal to 64. This tells us that the quadrilateral is inscribable by a circle, so the four points are concyclic, and you can enjoy a nice long walk on a circular path that visits all the attractions and ends back at the parking lot.
Concyclic points are points that lie on the same circle. We can prove that a given set of points are concyclic using various theorems based on how many points are in the set.
- Two points: Any set of two points are concyclic.
- Three points: Any set of three points that are non-colinear (don't lie on the same line) are concyclic.
- Four points: Ptolemy's Theorem states that a quadrilateral is inscribable in a circle if and only if the product of the measures of its diagonals are equal to the sum of the products of the measures of the pairs of opposite sides. Therefore, a set of four points is concyclic if it satisfies the properties of this theorem.
Thanks to these theorems, proving a set of 2, 3, or 4 points is made fairly easy, so it's a great idea to put these to memory. Meanwhile, enjoy your beautiful day at the park!
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