648. It is required to construct the circumference, having given three points in it.

649. It is required to construct a circumference which shall pass through the vertices of a given triangle.

650. It is required to find the locus of the centre of the circumference which shall pass through two given points.

651. With a given radius, it is required to construct the circle which shall pass through two given points.

652. It is required to construct the isosceles triangle, having given the base and the vertical angle.

653. It is required to construct a circumference which shall be a given distance from three given points.

654. It is required to construct a circle which shall have its centre in a given straight line and circumference passing through two given points.

655. It is required to find a point which shall be equally distant from two given points, and at a given distance from a third given point.

656. It is required to construct the equilateral triangle, having given one side.

657. It is required to trisect a right angle.

658. It is required to find a point which shall be equally distant from two given points, and also equally distant from two given parallel lines.

659. It is required to find a point which shall be equally distant from two given points, and also equally distant from two given non-parallel lines in the same plane.

660. It is required to find a point which shall be equally distant from two given parallel lines, and also equally distant from two non-parallel lines in the same plane.

662. It is required to find a point in one side of a triangle which shall be equally distant from the other two sides.

663. It is required to find a point which shall be equally distant from two non-parallel lines in the same plane, and at a given distance from a given point.

664. It is required to construct the right triangle, having given the two legs.

665. It is required to construct the right triangle, having given the hypothenuse and one acute angle.

666. It is required to construct the right triangle, having given one leg and adjacent acute angle.

667. It is required to construct the right triangle, having given one leg and the acute angle opposite.

668. It is required to construct the right triangle, having given the hypothenuse and one leg.

669. It is required to construct a tangent to a given circle, having given the point of contact.

670. It is required to construct a tangent to a given circle, which shall pass through a given point outside the circle.

Sug. Connect the centre of the given circle with the given exterior point. On this line as a diameter construct a circle, and join the points of its intersection of the given circle with extremities of the diameter.

671. It is required to construct the parallelogram, having given two sides and their included angle.

672. It is required to construct a circle within a given triangle, so that the sides of the triangle shall be tangents of the circle.

Sug. Consult Theorems 117 and 125.

673. It is required to construct the circle, having given a chord and angle made by the chord and a tangent.

Let AB be the given chord, and C the angle made by the chord and tangent.

the point of contact. Take DH = AB.

Then one extremity of the chord must be

Construct angle DHN equal to angle C.

Then H is the point of contact, and NH the tangent.

Find the locus of the centre of the circumference passing through D and H. (Problem 650.)

Then consult Theorem 202.

Hence T must be the centre of the circle required.

If from any point in the arc DOH (call the point Q) lines be drawn to D and H, how will the angle Q compare in magnitude with the angle DHN?

What is formed by the lines DH, DQ, and HQ?

If we call DH the base, what would you call the angle Q? What the point Q?

What might the arc DQH be named, then?

Q.E.F.

674. Having given the base and vertical angle of a triangle, it is required to construct the locus of its vertex. Sug. Consult the previous problem.

675. It is required to construct the triangle, having given the base, the vertical angle, and the altitude.

Sug. Use previous problem.

676. It is required to construct the triangle, having given the base, the vertical angle, and the median.

OPTIONAL PROBLEMS FOR ADVANCE WORK.

677. It is required to construct the triangle, having given the base, vertical angle, and perpendicular from one extremity of the base to the opposite side.

678. It is required to construct the isosceles triangle, having given the altitude and one of the equal angles.

679. It is required to construct the chord in a given circle, having given the middle point of the chord.

680. It is required to construct a circle whose circumference shall pass through the vertices of a given rectangle.

681. It is required to construct a tangent to a given circle which shall be parallel to a given straight line.

682. It is required to construct a tangent to a given circle which shall be perpendicular to a given straight line.

683. It is required to construct a tangent to a given circle which shall make a given angle with a given straight line.

684. It is required to construct the isosceles triangle, having given the vertical angle and a point in the base, in position.

685. It is required to construct the triangle, having given the altitude, the base, and an adjacent angle.

686. It is required to construct the triangle, having given the altitude, the base, and an adjacent side.

687. It is required to construct a rhombus, having given its base and altitude.

688. It is required to construct the triangle, having given the altitude and the sides which include the vertical angle.

689. It is required to construct the triangle, having given the altitude and angles adjacent to the base.

690. It is required to construct an isosceles triangle which shall have its vertical angle twice the sum of its other two angles.

691. It is required to construct the square, having given its diagonal.

692. It is required to construct the right triangle, having given the hypothenuse and perpendicular from the vertex of the right angle to the hypothenuse.

693. It is required to construct the locus of the centre of the circle of given radius tangent to a given straight line.

694. It is required to construct the circle of given radius which shall be tangent to two given non-parallel lines.

695. It is required to construct the circle of given radius which shall be tangent to a given straight line, and whose centre shall be in a given line not parallel to the former.

696. It is required to construct the circle which shall be tangent to a given line, and whose circumference shall pass through a given point.

697. It is required to find the locus of the centre of the circle of given radius which shall be tangent externally to a given circle.