I don't really want to copy-paste other people's work, so whatever I'll share here, is my own effort. For some cool properties, make sure to check out this, this and this

The following Definition will be carried thoughout the blog!
Define:
(Find out more about Ex-Points in math_pi_rate's blog, linked above)




Pre-Eliminaries

is defined as the foot of the perpendicular from to the -median
is defined as

Lemma 1: (Well Known) lies on

Proof: Let be the mid-point of , then, is a cyclic quadrilateral,
Perform inversion around with radius , Then, is a cyclic quadrilateral (Nine-Point Circle WRT , Hence, is also cyclic, indeed

Lemma 2: (Well Known) is the Intersection(Second) of and (Define and as the circles passing thorugh and tangent to at and )

Proof: Let and Let & , therefore, & ,

SubLemma: Let and let be the tangent to both the circles at , respectively, then bisects

Proof: It's known that, is the radical axis of , and let is the mid-point of

Back to the Proof (Lemma2) The above sublemma applied to & lies on the median and from , we have that,

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Main (Basic) Properties

Property 1: inversion () and is intersection of tangents at and to ,then,

Proof: It's known that swaps & , if & are the tangents to & , then and , and so by (Lemma 2) we're done!

Note: Generalization:
Proof: It directly comes from (Property 1)

Property 2: (math_pi_rate)

Proof: Since, is the orthocenter of , applying Brocard's Theorem on , we get that is the orthocenter of

Note: Generalization: Points are con-cyclic

Property 3: Let be the mid-point of , then, Points are concyclic

Proof: Apply Brocard's Theorem, on , we get that is the orthocenter of , and note that Points are concyclic

Property 4: lie on the same line

Proof: Pretty trivial to see that are concurrent at , hence, and are perspective triangles from , hence, by Desargues' Theorem on and , lie on the same line

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For the Application: check out the above mentioned links, given very nicely over there