Lines of Symmetry for a Triangle and its Circumcircle
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Exercise 5.1
1. $\triangle ABC$ is an equilateral triangle and circle $O$ is its circumcircle. How many lines of symmetry does figure 5.5 have?
This question includes visual content: The image shows Figure 5.5, which consists of a circle labeled with center 'O'. Inside this circle, there is an inscribed equilateral triangle labeled ABC. Vertex C is at the top, and vertices A and B are at the bottom left and bottom right respectively. The circle passes through all three vertices of the triangle.
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In this problem, we are asked to find the number of lines of symmetry for an equilateral triangle inscribed within its circumcircle O.
Lines of Symmetry Analysis
Let's start by reconstructing the figure. An equilateral triangle has three equal sides and three equal angles, and its circumcircle passes through all three vertices.
A line of symmetry is a line that divides a figure into two identical mirror images. For an equilateral triangle, these lines pass through each vertex and the midpoint of the opposite side.
Because the circle is perfectly centered around the triangle, any line of symmetry of the equilateral triangle will also be a line of symmetry for the circle. Here is the first line passing through vertex C.
1. Line through vertex C and center O.
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