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If A(x1,y1),B(x2,y2) and C(x3,y3) are th...

If `A(x_1,y_1),B(x_2,y_2) and C(x_3,y_3)` are the vertices of traingle ABC and `x_(1)^(2)+y_(1)^(2)=x_(2)^(2)+y_(2)^(2)=x_3^(2)+y_(3)^(2)`, then show that `x_1 sin2A+x_2sin2B+x_3sin2C=y_1sin2A+y_2sin2B+y_3sin 2C=0`.

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To prove that \( x_1 \sin 2A + x_2 \sin 2B + x_3 \sin 2C = 0 \) and \( y_1 \sin 2A + y_2 \sin 2B + y_3 \sin 2C = 0 \), we start with the given condition that the vertices of triangle \( ABC \) are equidistant from the origin. ### Step-by-step Solution: 1. **Given Condition**: We know that: \[ x_1^2 + y_1^2 = x_2^2 + y_2^2 = x_3^2 + y_3^2 = r^2 ...
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