Why is the distance from orthocenter to vertex twice the distance from circumcenter to opposite side?

Question

In the diagram above, $$2SP=AO$$ in description :

line from orthocenter is 2 times of line from circumcenter.

But I remember, someone in MSE said It's Euler line (I have read Wikipedia article but that doesn’t my question). My question is why line from the orthocenter is 2 times of line from the circumcenter?

Here, S is circumcenter, G is centroid and O is orthocenter.

Book says :

In $\triangle ABC$, the distance from the orthocenter O to A is OA, and the distance of the opposite line "BC" from the circumcenter S to the top of A is SP. $\therefore 2SP=AO$

Answer 1

1. Consider the following redrawn figure: remarkable points $H,G,O$ are aligned on the "Euler line" with

$$\vec{GH}=-2 \vec{GO}\tag{1}$$

(property also due to Euler). As a consequence triangle $GHC$ is the image of triangle $GOM_c$ by the homothety with center $G$ and ratio $-2$, giving the result.

2. Another kind of proof which does not use (1). Here is for example a copy of the classical book "College geometry" by Nathan Altshiller-Court. The proof relies on the construction of a diameter $CL$, allowing to say that $\angle LBC$ is a right angle, then spotting a parallelogram.)

(see as well here)

3. A completely different type of proof is based on Hamilton theorem (see page 2 here).

$$\vec{OH}=\vec{OA}+\vec{OB}+\vec{OC}$$

See here.