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If A, B, C are the centres of three circ...

If A, B, C are the centres of three circles touching mutually externally then the radical centre of the circles for `triangle`ABC is

A

centroid

B

orthocentre

C

circum centre

D

incentre

Text Solution

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The correct Answer is:
To solve the problem, we need to find the radical center of three circles that touch each other mutually externally, with centers A, B, and C. Let's go through the solution step by step. ### Step 1: Understand the Configuration We have three circles centered at points A, B, and C. Each circle touches the other two circles externally. This means that the distance between any two centers (A and B, B and C, C and A) is equal to the sum of their respective radii. ### Step 2: Draw the Circles and Triangle Draw the three circles centered at points A, B, and C. Label the points where the circles touch each other. The centers A, B, and C will form a triangle. ### Step 3: Identify the Radical Axes The radical axis of two circles is the locus of points that have the same power with respect to both circles. For our circles: - The radical axis of circles centered at A and B is the line that is perpendicular to the line segment AB and passes through the midpoint of AB. - Similarly, find the radical axes for circles centered at B and C, and C and A. ### Step 4: Locate the Radical Center The radical center is the point where the three radical axes intersect. Since the circles touch each other externally, the radical center will also have equal power with respect to all three circles. ### Step 5: Determine the Nature of the Radical Center In the case of three circles that touch each other externally, the radical center coincides with the incenter of triangle ABC formed by the centers of the circles. The incenter is the point where the angle bisectors of the triangle intersect and is equidistant from all sides of the triangle. ### Conclusion Thus, the radical center of the circles with centers A, B, and C is the incenter of triangle ABC. ### Final Answer The radical center of the circles is the **incenter** of triangle ABC. ---
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