# Construct Triangle Inscribed in a Circle

## How to Construct a Triangle Inscribed in a Circle?

In this chapter, we will help you learn how to construct an equilateral triangle inscribed in a circle with the help of a compass and a ruler or straightedge. You will also get familiar with the construction of the largest equilateral triangle that will fit in the circle, having each vertex touching the circle. This is quite the same as the construction of an inscribed hexagon, besides that we use every other vertex instead of all six.

### How to Draw Equilateral Triangle in a Circle with a Compass and Ruler?

Here, we are not acquainted with the information about the circle. That is we don’t know the diameter of the circle (and thus the radius) of the circle. We are also not aware of where the centre is. However, we have a compass and a ruler or straightedge in hand. Now, follow the suggested steps below and you will get your triangle inscribed in a circle.

Step I: Use the ruler to construct two chords on the circle.

Step II: Draw a perpendicular bisector to the chords using the compass. This perpendicular bisector must meet at the centre of the circle.

Step III: Make a line from the centre to one of the farthest points of one of the chords.

Step IV: Take the help of the compass in order to draw two lines which form a 30 degree with the radius drawn by step 3 and on the opposite side.

Step V: Stretch out these two lines to make two chords on the circle. These two chords therefore make an angle of 60 degrees to each other.

Step VI: Connect the other extreme of the two chords made by step 4. This together with the two chords makes the needed equilateral triangle. ### How to Construct Triangle Inscribe Given One Length

Using 3 methods, we will be performing constructions of an equilateral triangle given the length of one side, and the remaining two will be to draw an equilateral triangle inscribed in a circle.

Method 1:

Given: one side length measurement of the triangle.

Construct: an equilateral triangle.

Steps to Follow:

1. Set your compass point on A and calculate the distance to point B. Swing an arc of this size below or above the line segment.

2. Without disturbing or altering the span on the compass, place the compass point on B and swing the same arc, bisecting with the 1st arc.

3. Label the point of bisection as the 3rd vertex of the equilateral triangle.

### Proof of Construction

Circle A is congruent to circle B, seeing that they were each created using the same length of radius, AB. Because AB and AC are length measurements of radii of circle A, they are equivalent to one another. In the same manner, AB and BC are radii of circle B, and are equivalent to one another. Thus, AB = AC = BC by method of substitution (or transitive property). Since congruent line segments consist of equal lengths, equal segments and ΔABC are thus equilateral (containing 3 congruent sides).  Method 2:

Modification of the construction of a regular hexagon inscribed in a circle.

Given: A piece of paper

Construct: An equilateral triangle inscribed in a circle.

Steps to Follow:

1. Place your drawing compass to construct a circle; make sure to keep the compass span.

2. Put a dot, marked as A anywhere on the circumference of the circle to play a part of an initial point.

3. Without altering the span on the compass, establish the compass point on A and swing a small arc going through the circumference of the circle.

4. Without altering the span on the compass, shift the compass point to the bisection of the previous arc and the circumference and draw another small arc on the circumference of the circle.

5. Repeat this process of "stepping" around the circle till the time you return to point A.

Beginning at A, join every other arc on the circle to create the equilateral triangle. Q1. How to Draw an Equilateral Triangle?

Answer: This technique enables you to control the size of your equilateral triangle, since you construct it from scratch. Now follow the steps below:

1. Place your drawing compass to construct a circle; we'll call it Circle E.

2. Use your straightedge or ruler for the purpose of establishing a diameter (a line segment across the center Point E having endpoints on the circle itself). Label the endpoints of this construction as Points B and R.

3. Now, without disorganizing the compass, establish the needle arm down on Point R and swing around another complete circle.

4. Where the new circle bisects the old circle at two points below and above the diameter BRBR, label those as Points A Points S.

5. Connect Point B to Point A, and Point A to Point S and Point A to Point B. observe that you have drawn an equilateral triangle!

Q2. How to Prove the Construction of an Equilateral Triangle.

Answer: You can prove the triangle is equilateral inside the circle using your basic comprehension that any angle inscribed in a semicircle is a right angle, giving you right angles ∠BAR and ∠BSR, which forms two congruent right-angled triangles △BAR and △BSR sharing a common hypotenuse, BR.

This eventually makes AB ≅ BS (corresponding sides of congruent triangles), which also implies △BAS is isosceles. That also creates ∠BAS and ∠BSA congruent (base angles of an isosceles triangle are in congruence). We know ∠ABS measures 60°, leaving the two remaining angles to sum up to 120°, so they are each 60°. All the 3 interior angles of △BAS are equal, thus making it an equilateral triangle.