Euler Line theoram - part of the proof is not clear to me

[musicgold]

This is not homework. I am reading a book: "The art of infinite: The Pleasure of Mathematics" and pages 119-120 give a proof of the Euler Line theoram: the circumcenter, centroild and orthocenter of a triangle are always colinear (see the attached files).

1. Homework Statement
Page 119 shows a triangle with tree points. O, G, and H* are colinear. O is the circumcenter and G is the centroid of the triangle. H* is a point we hope to prove to be the orthocenter (H) of the triangle.

I am not clear on one point in the proof. I have put a question mark against the underlined part on page 120.
I am not sure how we can concldue that CK is parallel to OD (and therefore parpendicular to AB).

Homework Equations

The Attempt at a Solution

While I can see that $\Delta$ DOG and $\Delta$CH*G are similar, but I am not sure how we can jump to the concusion that CH* and OD are parallel. What am I missing?

 

[BvU]

Hi,
If you agree that the two triangles are similar, $\angle CH^* G = \angle DOG$ and there is a theorem: angles equal $\Leftrightarrow$ lines parallel

 

[musicgold]

Hi,
there is a theorem: angles equal $\Leftrightarrow$ lines parallel

Thanks.
Could you please point me to that theorem?

I know that if we start with two parallel lines and add two transversal lines, we get two similar triangles. However, here we are concluding that the lines are parallel. Can we not have have two similar triangles where neither of the sides are parallel?

For example, the solution of problem 4 on the following page has two similar triangles but none of the lines are parallel. What am I missing?

 

[BvU]

https://www.themathpage.com/aBookI/propI-27-28.htm

What am I missing?

Order of angle points/lines ?
In 'solution to problem 4' clearly $\angle BAC \ne \angle A'B'C$