Orthocentr and circumcentre formula in 3D

Question

Given three points in a 3D coordinate system, how can we find the orthocentre and circumcentre of the triangle formed by these points?

Answer 1

Given three Points: $$P,Q,R$$ Calculate: $$\begin{array}{l} {M_{PQ}} = \frac{1}{2}\left( {P + Q} \right)\\ {M_{QR}} = \frac{1}{2}\left( {Q + R} \right)\\ {M_{RP}} = \frac{1}{2}\left( {R + P} \right) \end{array}$$ and solve $$P + r \cdot {M_{QR}} = Q + s \cdot {M_{RP}}$$ for circumcentre. With $$\begin{array}{l} u = \overrightarrow {PQ} = Q - P\\ v = \overrightarrow {PR} = R - P\\ {u^ \bot } = u - \frac{{u \cdot v}}{{v \cdot v}}v \end{array}$$ and $$\begin{array}{l} a = \overrightarrow {QR} = R - Q\\ b = \overrightarrow {QP} = P - Q\\ {a^ \bot } = a - \frac{{a \cdot b}}{{b \cdot b}}b \end{array}$$ solve: $${M_{PQ}} + r \cdot {u^ \bot } = {M_{QR}} + s \cdot {a^ \bot }$$ for orthocentre.