triangle and two circles theorem

[Delong]

Can someone help me understand why this is the case? I tried forming a cartesian equation for the two circles but there were too many variables that it would be too messy to compute. Otherwise I am rather stuck on how to do it. I would appreciate it if someone can explain how to prove this theorem in accessible terms. Thanks.

In a triangle the inscribed circle touches the circle that passes through the midpoints of the sides.

[Studiot]

I have no idea what the 'two circles theorem' is but you seem to be discussing the incircle for a triangle.

The inscribed circle for any polygon is the circle to which each side is tangent.

To see why the tangent point is the mid point of each side look at my first sketch for some preliminary results.

From any point P outside any circle, centre O, two tangents may be drawn meeting the circle at M and N.
Since OM and ON are radii and NP and MP are tangents \hat{ONP} and \hat{OMP} are right angles.
Since OP is common to both triangles OMP and ONP they are congruent.

Therefore \hat{NPO} = \hat{MPO}
Therefore OP bisects \hat{NPM} and the two tangents
Therefore OP bisects any line such as MN, crossing PM and PN produced

Taking this result into my second diagram you can see that the line from each vertex of triangle ABC to the centre of the incircle produced bisects the vertex angle and the opposite side.
Further these lines meet the opposite sides at right angles, since OF, OG, OH are radii and the sides are tangents.
Further these lines meet at the centre of the incircle.

go well

[AlephZero]

See http://mathworld.wolfram.com/TangentCircles.html

Specifically, the diagram after equation 16.

[Delong]

So yes I will read these very soon. Thanks!