740. All regular polyedrons of the same number of faces are similar solids.

741. The intersection of the surfaces of two spheres is the circumference of a circle whose plane is perpendicular to the line which joins their centres.

742. Through any four points not in the same plane one sphere may be made to pass, and only one.

SUG's.-The four points may be considered as the vertices of a tetraedron Conceive perpendiculars drawn to the triangular faces from the intersections of lines drawn in these faces perpendicular to the sides at their middle points. These perpendiculars will meet at a common point (?), which is the centre of the circumscribed sphere (?).

[The student should show why only one sphere can be circumscribed.]

743. COR. 1.-The four perpendiculars erected at the centres of the circles circumscribing the faces of a tetraedron intersect at a common point.

744. COR. 2.-The six planes, perpendicular to the six edges of a tetraedron at their middle points, intersect at the centre of the circumscribed sphere.

745. One sphere and only one may be inscribed in any tetraedron. SUG.--Bisect the diedrals with planes.

746. The angle included by any two curves intersecting on the surface of a sphere, is equal to the angle included by the arcs of two great circles passing through the point of intersection, and whose planes produced include the tangents to the curves at their intersection.

748. To pass a circumference through three points, not in the same straight line, when the radius is so long as to render the ordinary method impracticable.

SUG.-Let A, B, and C be the three points; then are M and N other points in the same circumference.

M

Ε βλα

K

FIG. 410.

N

FIG. 409.

749. From two given points on the same side of a line given in position, to draw two lines which shall meet in that line and make equal angles with it.

SUG.-If a and B are equal, what is the relation of ME to EF?

750. To construct an isosceles triangle with a given base and vertical angle.

SUG.-See Prob. 4, p. 102.

751. To trisect a right angle.

SUG.-What is the value of an angle of an equilateral triangle?

752. Given the perpendicular of an equilateral triangle, to construct the triangle.

753. Given the diagonal of a square, to construct it.

754. To construct an isosceles triangle, so that the base shall be a given line, and the vertical angle a right angle.

it at I and of Drill

755. Given the sum of the diagonal and a side

of the square, to construct it.

756. To construct a triangle when the altitude, the vertical angle, and one of the sides are given.

ACT ABT (A gi

757. To construct a triangle when the sum of the three sides and the angles at the base are given.

758. In a right angled triangle the perimeter, and the perpendicular from the right angle upon the hypotenuse being given, to construct the triangle.

which shall meet in the same point of the line.

of a given line, to draw two equal straight lines

759. From two given points on the same side

M.

760. To pass a circumference through two given points, which shall have its centre in a given line.

P
FIG. 414.

761. To construct a quadrilateral when three sides, one angle, and the sum of two other angles are given.

SUG's. What is the fourth angle? When two sides and their included angle are known, there will be two cases, according as the two angles whose sum is known are adjacent to each other or opposite. In the latter case we have to describe a segment on a diagonal, which will contain the fourth angle. For the third case see Ex. 13, page 136.

762. To construct a quadrilateral when three angles and two opposite sides are given.

763. To bisect a trapezoid by a line drawn from one of its angles.,

764. In a given circle, to inscribe a triangle equiangular with given triangle.

SUG.-How does an angle at the centre compare with one inscribed in the same segment?

765. To describe three circles of equal diameters which shall touch each other.

766. In an equilateral triangle, to inscribe three equal circles which shall touch each other and the three sides of the triangle.

767. To describe a circle of given radius touching the two sides of a given angle.

SUG.-How far is the centre from each line?

768. To describe a circumference which shall be embraced between two parallels and pass through a given point within the parallels.

SUG.-In what line is the centre? How far from the given point?

769. To describe a circle with a given radius, which shall pass through a given point and be tangent to a given line.

770. To find in one side of a triangle the centre of a circle which shall touch the other two sides.

771. Through a given point on a circumference, and another

given point without, to describe a circle touching the given circumference.

SUG.-Consider in what two lines the centre must lie.

772. In the diameter of a circle produced, to determine the point from which a tangent drawn to the circumference shall be equal to the diameter.

SUG.-What is the relation between the radius, the required tangent, and the distance from the centre to the intersection of the produced diameter and the required tangent?

773. To describe a circle of given radius, touching two given circles.

774. In a given circle, to inscribe a right angle, one side of which is given.

775. In a given circle, to construct an inscribed triangle of given altitude and vertical angle.

" 776. To inscribe a square in a given rightangled isosceles triangle, one side being in the hypotenuse.

FIG. 417.

777. To inscribe a square in a given quadrant of a circle, the vertex of an angle being at the centre.

778. To find the centre of a circle in which two given lines meeting in a point shall be a tangent and a chord.

779. To describe a circumference which shall pass through a given point and be tangent to a given line at a given point.

780. To bisect a quadrilateral by a line

drawn from one of its angles.

SUG.-The demonstration is based upon the principle that triangles having equal bases and equal altitudes are equivalent.

FIG. 418.