102. Find a point, the sum of whose distances from three given points may be a minimum. 103. A line drawn through the intersection of two tangents to a circle is divided harmonically by the circle and the chord of contact. 104. To construct a quadrilateral similar to a given one whose four sides shall pass through four given points. 105. To construct a quadrilateral, similar to a given one, whose four vertices shall lie on four given lines. 106. Given the base of a triangle, the difference of the base angles, and the rectangle of the sides; construct the triangle. 107. ABCD is a square, the side CD is bisected in E, and the line EF drawn, making the angle AEF = EAB; prove that EF divides the side BC in the ratio of 2 : 1. 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. 117. The product of the bisectors of the three angles of a triangle whose sides are a, b, c, is 118. In the same case the product of the alternate segments of the sides made by the bisectors of the angles is 119. If three of the six points in which a circle meets the sides of any triangle be such, that the lines joining them to the opposite vertices are concurrent, the same property is true of the three remaining points. 120. If a triangle A′B′C′ be inscribed in another ABC, prove is equal twice the triangle A′B′C′ multiplied by the diameter of the circle ABC.