108. If any chord be drawn through a fixed point on a diameter of a circle, and its extremities joined to either end of the diameter, the joining lines cut off, on the tangent at the other end, portions whose rectangle is constant.
109. If two circles touch, and through their point of contact two secants be drawn at right angles to each other, cutting the circles respectively in the points A, A′; B, B′; then AA′2 + BB′2 is constant.
110. If two secants at right angles to each other, passing through one of the points of intersection of two circles, cut the circles again, and the line through their centres in the two systems of points a, b, c; a′, b′, c′ respectively, then ab : bc :: a′b′ : b′c′.
111. Two circles described to touch an ordinate of a semicircle, the semicircle itself, and the semicircles on the segments of the diameter, are equal to one another.
112. If a chord of a given circle subtend a right angle at a given point, the locus of the intersection of the tangents at its extremities is a circle.
113. The rectangle contained by the segments of the base of a triangle, made by the point of contact of the inscribed circle, is equal to the rectangle contained by the perpendiculars from the extremities of the base on the bisector of the vertical angle.
114. If O be the centre of the inscribed circle of the triangle prove
115. State and prove the corresponding theorems for the centres of the escribed circles.
116. Four points A, B, C, D are collinear; find a point P at which the segments AB, BC, CD subtend equal angles.