Here are some more basic properties!

Define:


Credits to math_pi_rate for giving such amazing names



Pre Eliminaries

is defined as
is defined as
Note: This implies that lie on the same circle
Also Define:
as the Radical Center of
Similarly, define and

Lemma 1: lie on the same line (This line is known as the orthic axis)

Proof: Note, , hence, is the center of perspectivity of and , applying Desargues' Theorem on perspective triangles and implies lie on the same line
Note(1): Infact the above proof follows from the fact that the Orthic Axis is defined as the Axis of Perspectivity of the orthic triangle and reference triangle
Note(2): Another way of defining the Orthic Axis, Let the reflections of over be , Denote, as the midpoints of , then the polar of WRT is the Orthic Axis WRT

Lemma 2: is the diameter of

Proof: Easy to see that finishes the lemma
Note: If is the midpoint of , then,

Lemma 3: Let be any point on , and Let and , then,

Proof: Let and , then,

Lemma 4: Let intersect at , then is isosceles triangle

Proof: It's Well-Known that , then using (Lemma 3) and the fact that is cyclic trapezoid is isosceles

Generalization(Lemma 3): Let intersect at , then, is the angle bisector of
Proof: Follows from (Lemma 4)
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Main (Basic) Properties

Property 1: is the foot of altitude from in OR (Restatement) lies on

Proof: Let be the antipode of WRT , hence, it easily follows that and then, Notice that is the center of ,and finish off with Brokard's Theorem
Consequence(1): From (Property 1), we can show that, lies on
Consequence(2): From (Property 1), is a cyclic quadrilateral

Property 2: Let the reflections of over be , Denote, as the midpoints of , then the polar of WRT is the Orthic Axis WRT

Proof:(enhanced) Define: and let a parallel line () to through intersect at , Let , Applying (Lemma 4) lies on the polar of WRT , also it is trivial to see that, and the fact that the polar of WRT and are parallel, now take homothety() centered at , such, , then, it's clear that the polar of WRT will be the Orthic Axis WRT

Property 3: The Euler line WRT is orthogonal to Orthic Axis WRT

Proof: Notice that, if, , then , then Apply (Property 2), the orthogonality

Property 4: lie on the same line

Proof: This follows from the definition of combined with (Consequence(1) from Property(1))

Property 5: Let , then, lies on

Proof: Let be the antipode of WRT , Hence, from (Property 1), , and then, it is easy to see, and using the fact that

Property 6: Finding Some Cyclic Quadrilaterals!
(1) Points lie on a circle
(2) Points lie on a circle
(3) Points lie on a circle

Note: Adding the ( Humpty Point), and
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And....of course, there are many more properties that I found, But I haven't listed them over here, because many of them are pretty trivial and kinda useless(not so interesting), So I will limit these properties so as not make this post boring