BOOK VI. APPLICATION OF THE THEORY OF PROPORTION (continued 2)

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.

121. Construct a polygon of an odd number of sides, being given that the sides taken in order are divided in given ratios by fixed points.

122. If the external diagonal of a quadrilateral inscribed in a given circle be a chord of another given circle, the locus of its middle point is a circle.

123. If a chord of one circle be a tangent to another, the line connecting the middle point of each arc which it cuts off on the first, to its point of contact with the second, passes through a given point.

124. From a point P in the plane of a given polygon perpendiculars are let fall on its sides; if the area of the polygon formed by joining the feet of the perpendiculars be given, the locus of P is a circle.