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Problem 599

# Inscribe an equilateral triangle In a given circle.

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To inscribe an equilateral triangle in a given circle, first, identify the center (O) and radius (r) of the circle. Draw a radius (OA) and construct a 60-degree angle at point A. Create a circle with center A and radius r, intersecting the original circle at points B and C. Finally, connect points A, B, and C to form the inscribed equilateral triangle.
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## Step 1: Find the center and radius of the given circle.

First, identify the center and radius of the given circle. Let the center be O and the radius be r.

## Step 2: Draw a radius on the circumference of the circle.

Draw a line segment from the center O to a point A on the circle. OA will serve as one side of the equilateral triangle. Since all sides of an equilateral triangle are equal, the length of OA is also r.

## Step 3: Determine the angles of the equilateral triangle.

An equilateral triangle has three equal angles of 60 degrees each. So, we will create a 60-degree angle at point A.

## Step 4: Construct a circle with radius r and center A.

Draw a circle with center A and radius r. This circle will intersect the given circle at two points, which will be the vertices of the equilateral triangle.

## Step 5: Find the intersection points of the two circles.

Label the intersection points of the two circles as B and C. These points will be the vertices of the equilateral triangle inscribed in the given circle.

## Step 6: Construct the equilateral triangle.

Draw the line segments AB and AC, forming an equilateral triangle with vertices A, B, and C. The equilateral triangle ABC is now inscribed in the given circle, with its vertices lying on the circumference of the circle.

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