S. L. Loney's Coordinate Geometry (1895 Ed.) Article 66, Section III

Question

I couldn't decipher the result mentioned in the second paragraph of section III (particularly due to the words "in which the origin lies") but based on the further elaboration by the author in square brackets [] i.e.,
The angles between the two lines & the angles between their perpendiculars from the origin are supplementary (i.e., their sum is 180$^\circ$).

And second thing I couldn't understand that is also mentioned in the square brackets is, "O, $R_1$, $R_2$ & A lie on a circle" $-$ How ?

I've tried to prove the supplementary angle result based on the theorems of Circle mentioned in the 10 standard math textbooks (last lesson taught in Euclidean geometry in most of the schools) but I failed.
Here's the image of the exact page
And if image link does't work, here's the archive direct link to the book page no. 43: https://archive.org/stream/elementsofcoordi00lone#page/43/mode/1up

So I've got two problems the one in bold above & the How question. Any help would be appreciated.

Answer 1

Hope the following picture can help.

The key factor is “$\angle OR_1L_1 + \angle OR_2L_2 = 180^0$” implies “$OR_1AR_2$” is a cyclic quadrilateral".

$\angle R_1AL_2$ being the exterior angle of that quadrilateral, we therefore have $\angle R_2OR_1 = \angle R_1AL_2 = \beta - \alpha$.

Answer 2

The beginning of Article 66 on page 42 gives the missing context that the two lines intersect in the point $A$. In order to understand Section III and the bracketed text, it is best to draw a new diagram with the point $O$ as one end and $A$ as the other end of a diameter of a circle. Any other point $R$ on the circle forms a right triangle with the diameter being the hypotenuse and the right angle being at $R$ and, thus, the line $OR$ is perpendicular to line $RA$. Here, the two lines intersect the circle at $R_1$ and $R_2$.

The two right triangle side angles are complementary since their sum plus the right angle add up to two right angles. If the points $R_1$ and $R_2$ are on opposite sides of the diameter $OA$, then the angle between $R_1O$ and $R_2O$ is supplementary to the angle between $R_1A$ and $R_2A$, which is easy to see. If they were of the same side, then the angle between the perpendiculars from $O$ is the same as the angle between the lines from $A$.