Math Olympiad Question?

Question

"Let BC be the diameter of the circle T with centre O. Let A be a point on T such that 0 < (
Circle T on the real question is Circle Gamma.

I'd really appreciate it, if you told me what to do whether than give me the answer.

I mean hints as to how to do it.

Answer 1

Triangle EFC is inscribed in the circle with the center O. There is a theorem about incenters of such triangles.


THEOREM.
If the bisector of the angle C intersects the circumcircle at point A then:
1. |OA| is perpendicular to |EF|, or A is the midpoint of arc EF.
2. The circle with the center A passing through E and F also passes through J, and it also passes through the so-called excenter.
3. The circumradius R and the inradius r satisfy the relation |OJ|^2 = R^2- 2rR (using this property is not required in your case).


If you do not know this theorem, then you need to partially re-invent it while solving the problem. This is tough, especially during the competition. If you know the theorem, then the task is significantly simplified.

Checking Point 1 is relatively easy. To check Point 2 you need to show that |AJ| = |AE| = |AF|. Consider quadrilaterals OEAF and ODAJ. Prove that one of them is a rhombus and another is a parallelogram, and you are there.

Answer 2

First of all realize that a triangle has three important internal points. [EDIT maybe four important internal points].

The centroid which is the meet of the lines from the vertices to the opposite midpoints.

The circumcentre, which is the centre of the circumcircle.

The incentre[EDIT, perhaps this is not the right name, there is also an incircle, for which the sides of the triangle are tangents], which is the meet of the perpendiculars to a side through the opposite vertex.

These points have many fascinating properties. I have no further information, I'm afraid.