Maths · Co-ordinate of the centroid, orthocentre and circumcentre of a triangle
is any point on the circumcircle of other than the vertices. is the orthocenter
\( P \) is any point on the circumcircle of \( \triangle A B C \) other than the vertices. \( H \) is the orthocenter of \( \triangle A B C, M \) is the mid- point of \( \boldsymbol{P H} \) and \( \boldsymbol{D} \) is the mid-point of \( B C . \) Then
- A. \( A P \) is opposite side of \( D M \)
- B. \( D M \) is parallel to \( A P \)
- C. \( D M \) is perpendicular to \( A P \)
- D. None of these
Step-by-step solution
Using vector analysis with circumcenter as origin, we have H = A + B + C. D = (B+C)/2, M = (P+H)/2 = (P+A+B+C)/2. Then DM = M - D = (P+A)/2 and AP = P - A. Their dot product is (A+P)·(P-A)/2 = (|P|^2 - |A|^2)/2 = 0 since both lie on circumcircle. Hence DM ⟂ AP.
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