Hello, xwrathbringerx!

Two circles touch internally at a point

and the smaller of the two circles passes through , the centre of the larger circle.

is any chord of the larger circle, cutting the smaller circle

The tangents to the larger circle at and meet at a point

Prove:

a) is bisected at

b) are collinear.

Draw

Draw radii

Tangents to a circle from an extrenal point are equal.

Points and are equidistant from points and

. . Hence, is the perpendicular bisector of

is a diameter of the small circle.

is inscribed in a semicircle: .

Hence, lies on

And the two proofs follow . . .