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
a) is bisected at
b) are collinear.
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 . . .