Answer: The circle is the incircle of the triangle
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Explanation:
Form line segment PO. Then find the midpoint of it, which I'll call point M.
Draw a circle centered at point M, and this circle will pass through points P and O. This second circle intersects the first circle at two new points which I'll call R and S. Refer to figure 1 in the diagram below. Points R and S are points of tangency for the tangent lines PR and PS respectively. The lines touch the circle at exactly one point.
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Now to find the tangent through point A. Use your compass to form a circle of radius AO, and this circle is centered at point A. Then draw ray OA. This ray intersects the new circle at point B. Next, you'll apply the steps needed to form the perpendicular bisector of segment OB. This perpendicular bisector is exactly the tangent line we need. Recall the tangent is always perpendicular to the radius at the point of tangency. This line tangent at point A is perpendicular to radius OA. Refer to figure 2.
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Figure 3 shows the combinations of figures 1 and 2. I've erased all of the dashed lines so that all we have are the triangle and the circle inside. This circle is known as the incircle. It's a special circle that is as large as possible, but does not spill outside the triangle. One possible application is that we could have say a triangular box (aka triangular prism) and we want to find the largest circle possible to pack in such a box.
