Define:




Some Relations

Relation 1:

Relation 2:

Proof: This follows from the fact that

Relation 3:

Proof: Easy to get that, And for,
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Lemmas

Lemma 1: In , if are the Isogonal Conjugates WRT to and if , , then

Proof: WLOG, assume, , then,

Lemma 2: Let , such, , Draw a parallel through to and let , then,

Proof: From (Relation 3),

Lemma 3: Let & , then, is the midpoint of

Proof: From (Relation 2),

Consequence: From (Lemma 2), it is quite evident, that, is

Lemma 4: is also the midpoint of and

Proof: Trivial and Quite evident from Consequence (Lemma3)

Lemma 5: Let (parallel line) , then, is midpoint of

Proof: This just follows as a consequence of (Lemma 2) and (Lemma 3)

Lemma 6: Let , and , then,

Proof: Apply Pascal's Theorem on , we already have, & , Hence,

Lemma 7: & are Isogonal Lines WRT

Proof: Direct Converse of (Lemma 1)

Lemma 8: is a cyclic quadrilateral

Proof: is cyclic
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(Taken from the configuration of IMO 2010/P2)