The original problem is:
If a, b, c, d are the position vectors of points A, B, C, D respectively such that $$(\vec{a}-\vec{d}). (\vec{b}-\vec{c})= (\vec{b}-\vec{d}). (\vec{c}-\vec{a})= 0$$then prove that D is the orthocentre of ${\Delta}$ ABC.
How do we go about proving that a point is the orthocentre of a triangle? I've tried expanding the dot product but I don't seem to get anywhere.
Best Answer
That formula states that $AD\perp BC$ (so that $D$ is on the altitude from $A$ to $BC$) and that $AC\perp BD$ (so that $D$ is on the altitude from $B$ to $AC$). As $D$ is on two altitudes of the triangle, it is its orthocentre.