If the orthocentre of the triangle formed by the lines and , is the centroid of another triangle, whose circumcentre and orthocentre respectively are and , then the value of is .

Question

If the orthocentre of the triangle formed by the lines  and , is the centroid of another triangle, whose circumcentre and orthocentre respectively are  and , then the value of  is .

Canvas Classes · Accepted Answer

🧠 Euler Line Hands You  in One Line

The line-triangle's orthocentre equals the second triangle's centroid, and Euler's relation says . Plug in  and  — they're scaled mirror images through origin, so . The whole problem reduces to: " is the orthocentre of the triangle from three given lines — find ."

🖼️ Visual Sketch

Three lines  forming a triangle, with origin sitting at the intersection of all three altitudes.

🗺️ The Clean Algebraic Route

: solving  and  gives .

The altitude from  is p

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