MC, K, T are collinear and MC K.MC T = MC B 2 = MC A2 2. For the sake of definiteness we assume that the circle is tangent to the side AB at K, to arc ACB at T, and CD is a cevian tangent to ω at L.
(Curvilinear Incircles) A curvilinear incircle ω is a circle tangent to an arc of (ABC) containing all three vertices of ABC and also tangent to one of its sides. 
∠AMA0 I = ∠IM B where M is the midpoint of BC. OI is the Euler line of the contact triangle. Antipode of A in (ABC), the foot of the D-altitude in the contact triangle and I lie on a line which passes through the second intersection of (ABC) and (AEIF ). Consequences include: The tangent to the incircle at AD ∩ (I) passes through X, AD bisects ∠B(AD ∩ IK)C. 
If DEF is the orthic triangle of 4ABC then is harmonic. The O is the orthocenter of the medial triangle. Reflections of the orthocenter over the sides and the midpoints of the sides lie on (ABC). We denote H as the orthocentre, O as the circumcentre, G as the centroid, N as the nine-point centre, I as the incentre, Na as the Nagel point and Ge as the Gergonne point of ∆ABC. By (XY Z) we denote the circumcircle of ∆XY Z, by (XY ) the circle with XY as diameter, and by (M, r) the circle with centre M and radius r, the radius being dropped when the context is clear. These will help you write some really elegant solutions (and will also help you to simplify your bashes in cases of some problems that don’t yield easily to synthetic solutions.) So have fun proving these lemmas and using them to the fullest advantage in your Olympiad journey! Usually these lemmas will be intermediate results that will help you reach the solution in many cases, and maybe even trivialize the problem. 
This list of lemmas is also intended to be a list of some easier problems and also as some configurations that frequently appear on contests. Here is a collection of some useful lemmas in geometry, some of them well known, some obscure and some by the author himself. Lemmas In Olympiad Geometry Navneel Singhal JGeometry is the art of correct reasoning from incorrectly drawn figures.
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