Constructing Circles in Geometry

In Euclidean geometry, a circle is a closed curve formed by the set of all points in a plane at a fixed distance (called the radius) from a given point (called the center). Given any three non-collinear points P, Q and R in the plane, there exists a unique circle passing through all three. This is known as the circle of Apollonius.

A method for constructing a circle is illustrated below.

First, we need to find the midpoint of line segment PQ. To do this, we draw a line segment PR perpendicular to PQ and then draw another line segment QR perpendicular to PQ such that PR = RQ. Points P, R and Q now form an equilateral triangle and since an equilateral triangle has equal sides, we know that segment PQ = segment PR = segment RQ. Therefore, the midpoint of segment PQ is point R.

Next, we need to construct a circle with center at point R and radius RP. To do this, we can use the compass and ruler postulate which states that given any two points A and B, there exists a circle with center A and radius AB. Therefore, using our compass, we set its width to RP and then place one end of the compass at point R. We then swing an arc above or below PQ until the other end of the compass meets the paper again. This then forms our desired circle with center R and radius RP. Finally, to complete the construction, we simply connect points P and Q with a straight line segment.

There you have it! A step-by-step guide to construct a circle given any three non-collinear points in the plane using only a straightedge and compass. Stay tuned for future posts where we will explore more advanced methods of constructing circles as well as other fundamental shapes in Euclidean

FAQ

How do you make a circle in geometry?

There are many ways to make a circle in geometry. Some of the most common methods include using a compass or ruler to draw a perfect circle, or using a protractor to measure and draw an accurate circle. You can also make a circle by drawing three non-collinear points and connecting them with a curve. Another method is to draw a polygon with many sides and then take the limit as the number of sides approaches infinity. This will result in a circle. Finally, you can use trigonometric functions to define a circle with any center and radius.

What are the steps of construction geometry?

There are a few different methods of construction geometry, but the most common is called Euclidean construction. This involves using a compass and straightedge to create geometric shapes. First, you use the compass to draw a circle with any given center and radius. Then, you use the straightedge to draw a line segment between any two points on the circumference of the circle. Next, you use the compass to bisect the angle formed by the line segment and the circle. This will create two new line segments of equal length. Finally, you use the straightedge to connect the ends of the line segments, creating a geometric shape.