The radical centre of three circles described | vedclass

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(A) Let the vertices of the triangle be $A, B,$ and $C$. Consider the three circles having the sides $AB, BC,$ and $CA$ as their diameters.The equatio

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The orthocentre

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(A) Let the vertices of the triangle be $A, B,$ and $C$. Consider the three circles having the sides $AB, BC,$ and $CA$ as their diameters.The equation of a circle with diameter endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$.The radical axis of any two of these circles is the common chord of the two circles. For circles described on the sides of a triangle as diameters,the radical axis of any two circles is the altitude of the triangle passing through the vertex common to the two sides.Since the radical centre is the point of intersection of the radical axes of the three circles taken in pairs,it is the point of intersection of the three altitudes of the triangle.Therefore,the radical centre is the orthocentre of the triangle.

The radical centre of three circles described on the three sides of a triangle as diameters is

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