Circle Passing Through Three Points Explained Clearly

The circle is a plane figure for which all the points pass through a single plane with equal distance. As it is a plane surface, it's a solid representation of a sphere. The circle has different terminology like radius, diameter, arc, chord circumference, etc. All the terms that arise in the concept of circle, if the circle passes through three points, what is the equation etc. will be studied now. 

Circle Passing through 3 Points

If we want to draw a straight line, we need one starting point and one ending point. That means a line needs two points to draw. Similarly, if we are supposed to draw a circle, we need to have some points. But, unlike line segments, we have different ways to draw a circle. Even if a circle can be drawn using a single point because it doesn't have multiple planes, the starting point will also become the ending point.

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Similarly, we can draw a circle using two points. As we had run several circles from a single point, we can also draw several circles from two points. But the current task is to draw a circle passing through three points.

While considering a circle passing through three points, it is important to observe two cases. Because the points may be either collinear or non-collinear, the circle may pass through collinear points or pass through non-colonial points.

The Circle Passes through Collinear Points

Collinear points mean the points which lie on the same line or in the same direction. So by considering these collinear points, if we draw a circle, then the result will be that the circle touches only two points and the third Point May observe either inside of the circle or outside of the circle. In this case, the circle never touches all three points.

The Circle Passes through Three Non-collinear Points.

For this, we need to draw two lines that need to touch the three points. Here it is important to draw two bisectors for two line segments. Now let's take the center and then draw a circle with the center of the circle passing through 3 points. Constituting the equal distance from the center to all sides of the circle, it is known as the radius of a circle passing through 3 points.

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By considering the center and radius of the circle passing through 3 points, it is easy to find the equation of the circle passing through 3 points. Also, the circle passing through collinear points can be stated by proof and theorem.

Find the Equation of the Circle Passing through 3 Points 

It is very easy and simple to find the equation of a circle passing through 3 points. For this, we need three non-collier parts with which the circle passes through them. 

So let us consider three points initially.

P (\[x_{1}, y_{1}\]), Q (\[x_{2}, y_{2}\]), and R (\[x_{3}, y_{3}\]) are the three points. So, we need to get an equation of a circle passing through these 3 points.

We have a general equation using the two variables.

It is ,\[x^{2} + y^{2} + 2gx + 2fy + c = 0\]

Like the General equation, we need to write equations for three variables with which the circle has to pass through them. Then the obtained equations will be,

\[x{_{1}}^{2} + y{_{1}}^{2} + 2gx_{1} + 2fy_{1} + c = 0\]

\[x{_{2}}^{2} + y{_{2}}^{2} + 2gx_{2} + 2fy_{2} + c = 0\]

and \[x{_{3}}^{2} + y{_{3}}^{2} + 2gx_{3} + 2fy_{3} + c = 0\]

Now, we need to find out the values of g, f, c by solving the three equations of three variables. After getting those values, substitute them in the general equation. The general equation itself is the required equation of the circle passing through three given points. 

After getting all the values, we need to draw a circle from that circle. We can easily find the radius of the circle that passes through three points and the center of the circle passes through three points. It is also remembered that the radius should be equal from the center to any point of the circle.

Hence it is the process and explanation of finding the equation of a circle that passes through three points. Those three points should be made collinear because we had already gone through the concept of collinear and non-collinear points.

Creating a 3-Point Circle using a Compass and Straightedge

• Join the points to form two lines

• Create a perpendicular bisector of one line

• Create a perpendicular bisector of another line

• Where they cross is the center of the circle

• Position the compass in the center, adjust the length to any point, and then draw your circle!

Some Important Key Points from Circles:

• Circle: A collection of all points in a plane at a fixed distance from a fixed point. The fixed point is called the center and the fixed distance is called the radius.

• Chord: Any part of a line that combines two points in a circle is called a chord. The chord that passes through the center of the circle is called the width.

• Secant: A line that crosses a circle by two points is called a secant.

• Tangent: A line that crosses a circle only in one place.

• Width: The width is twice the radius. It is the longest voice in the circle that passes through the center. All diameters are the same length.

• Circular: The circumference of a circle is called the circumference of the circle.

• Sector: The area formed by the arc and the two circular radii, by connecting the center to the end of the arc, is called the Field.

• Theorem 1: The tangent of any circle point is perpendicular to the radius through the contact area.

• Theorem 2: The length of the tents drawn from the outside to the circle is equal.

Some Preparation Tips for Math Students Studying Circles

Statistics is not as challenging as it sounds. For many, it is a matter of dread. However, it is a lesson that will help you even after you leave school! It is also widely used in other subjects such as Physics and Chemistry. The great thing about Math is that, once you get it, it can be your most scoring lesson. Impossible, it is easier than it looks.

Practice as Much as You Can 

Statistics are a hands-on topic. You can’t just ‘read’ the chapters, you have to understand the concepts and keep practicing.

Start by Solving Examples

Do not start by solving complex problems. If you have just understood the chapter, solving difficult Math will give you the wrong answer and discourage you. It may make you hate Math even more. Instead, start simple. Solve examples in your textbook.

Clear All Your Doubts

It is easy to cling to doubts in Math. Do not let your doubts form, remove them as soon as possible. The sooner you resolve your doubts, the better off you will be in those articles. Ask your class teacher, friends, or online about the app.

Not All Formulas

When you see something enough, it is registered in your memory, even if it is unconscious. That is why some people choose to attach drawings or formulas to their study table or their room. Make flashcards of all the formulas in your textbook and decorate your room with them, at least until the exam ends!

Understand the Origin

You may think that the release is not so important from a test perspective, but it is important for understanding. You cannot always read the formula, you need to understand the mind behind it.