13. The rectangle contained by the perpendiculars from any point in a circle, on the diagonals of an inscribed quadrilateral, is equal to the rectangle contained by the perpendiculars from the same point on either pair of opposite sides. 14. The rectangle contained by the sides of a triangle is greater than the square on the internal bisector of the vertical angle, by the rectangle contained by the segments of the base. 15. If through A, one of the points of intersection of two circles, we draw any line ABC, cutting the circles again in B and C, the tangents at B and C intersect at a given angle. 16. If a chord of a given circle pass through a given point, the locus of the intersection of tangents at its extremities is a right line. 17. The rectangle contained by the distances of the point where the internal bisector of the vertical angle meets the base, and the point where the perpendicular from the vertex meets it from the middle point of the base, is equal to the square on half the difference of the sides. 18. State and prove the Proposition analogous to 17 for the external bisector of the vertical angle. 19. The square on the external diagonal of a cyclic quadrilateral is equal to the sum of the squares on the tangents from its extremities to the circumscribed circle. 20. If a variable circle touch a given circle and a given line, the chord of contact passes through a given point. 21. If A, B, C be three points in the circumference of a circle, and D, E the middle points of the arcs AB, AC; then if the line DE intersect the chords AB, AC in the points F, G, AF is equal to AG. 22. Given two circles, O, O′; then if any secant cut O in the points B, C, and O′ in the points B′, C′, and another secant cuts them in the points D, E; D′, E′ respectively; the four chords BD, CE, B′D′, C′E′ form a cyclic quadrilateral. 23. If a cyclic quadrilateral be such that a circle can be inscribed in it, the lines joining the points of contact are perpendicular to each other. 24. If through the point of intersection of the diagonals of a cyclic quadrilateral the minimum chord be drawn, that point will bisect the part of the chord between the opposite sides of the quadrilateral. 25. Given the base of a triangle, the vertical angle, and either the internal or the external bisector at the vertical angle; construct it. 26. If through the middle point A of a given arc BAC we draw any chord AD, cutting BC in E, the rectangle AD.AE is constant. 27. The four circles circumscribing the four triangles formed by any four lines pass through a common point. 28. If X, Y , Z be any three points on the three sides of a triangle ABC, the three circles about the triangles Y AZ, ZBX, XCY pass through a common point. 29. If the position of the common point in the last question be given, the three angles of the triangle XY Z are given, and conversely.