there be two solutions, one solution, or no solution possible?

2. ABC is any triangle and CPQ is drawn perpendicular to the bisector of the angle A meeting it in P, and the side AB in Q. Prove that AQC is an isosceles triangle, and that the angle QCB is equal to half the difference of the angles ABC and ACB.

3. If a quadrilateral figure has all its sides equal and one angle a right angle, all its angles are right angles.

4. If the diagonals of a quadrilateral figure bisect each other, the figure is a parallelogram.

5. If a quadrilateral figure has two of its opposite sides parallel, and the other two sides equal but not parallel, any two of its opposite angles are together equal to two right angles.

6. The parts of all perpendiculars to two parallel lines intercepted between them are equal.

PART V.

TO EUCLID I. 34.

XXV.

1. If the line which bisects the vertical angle of a triangle also bisects the base the triangle is isosceles.

2. Prove that there is one and only one point which is equidistant from three given points not in the same straight line.

3. Show that four equal right-angled isosceles triangles can be arranged round one common vertex so as to form a square.

4. In the triangle ABC the side AB is bisected at M and MN is drawn parallel to BC to meet AC in N. Prove, by drawing NP parallel to AB, that MN bisects the side AC.

5. Any point P is taken in the base BC of an isosceles triangle, and PM and PN are drawn parallel to the equal sides AB, AC to meet them in M and N. that the sum of PM and PN is constant.

Prove

6. ABC is any triangle and CPQ is drawn perpendicular to the bisector of the angle A meeting it in P, and the side AB in Q. Prove that each of the angles AQC and ACQ is equal to half the sum of the angles ABC and ACB.

XXVI.

1. If through any point equidistant from two parallel straight lines, two other straight lines be drawn cutting the parallel straight lines, one in the points A and C, the other in B and D, prove that AC=DB.

2. Prove that there are four and only four points in a plane, each of which is equidistant from the three sides of a triangle.

3. Show that six equilateral triangles can be arranged round one common vertex so as to make a regular hexagon.

4. In the triangle ABC the two medians BY and OZ are drawn to intersect in O, and through C, CE is drawn parallel to BY. Join AO, and produce it to meet BC in X, and CE in E. Join BE. Prove that AE is bisected in O, that BOCE is a parallelogram, and that AX is the third median of the triangle ABC.

5. Draw a straight line through a given point, so that the part of it intercepted between two given parallels may be of a given length.

Draw AC

6. AB is a given finite straight line. making the angle BAC acute. Produce AC to D, and then to E, making CD and CE each equal to AC. Join BE, and draw CP and DQ parallel to BE. Prove that AB is trisected in the points P and Q.

XXVII.

1. Any line AX is drawn through the angle A of the parallelogram ABCD, and BP, CQ and DR are drawn perpendicular to AX. If C is the angle opposite A,

prove that CQ is equal to the sum or difference of BP and DR, according as AX falls without or intersects the parallelogram.

2. If two straight lines are parallel to two other straight lines, each to each, then the angles contained by the first pair are equal to the angles contained by the other pair.

3. Show that eight equal triangles can be arranged round one common vertex so as to form a regular octagon.

4. Bisect AC and AB the sides of the triangle ABC at the points Y, Z; and draw AP perpendicular to BC. Prove that the angle YPZ is equal to the angle BAC.

5. If two parallelograms have two adjacent sides of the one equal to two adjacent sides of the other, each to each, and one angle of one equal to one angle of the other, the two parallelograms are equal in all respects.

6. If the angle between two adjacent sides of a parallelogram be increased, but the lengths of the sides. remain the same, the diagonal through their point of intersection will be diminished.

XXVIII.

1. In a given straight line find a point which is equidistant from two given straight lines. When is this impossible?

2. Half the base of a triangle is greater than, equal to, or less than the line joining the vertex to the middle point of the base, according as the vertical angle is obtuse, right or acute.

3. Find the locus of a point which is at a given distance from a given straight line.

4. In a right-angled triangle ABC, having the right angle ACB, if the angle CAB is double of the angle ABC, then AB is double of AC.

5. Two right-angled parallelograms are equal if two adjacent sides of the one are equal to two adjacent sides of the other, each to each.

6. Bisect AB, CD, two opposite sides of a parallelogram ABCD at M and N. Join CM and NB. Prove that DM and NB trisect the diagonal AC

XXIX.

1. Prove by the method of superposition that if the four angles of a quadrilateral figure are all equal, its opposite sides are equal.

2. If in the sides AB, AC of the triangle ABC, in which AC is greater than AB, points D, E be taken so that BD and CE are equal, prove that CD is greater than BE.

3. Draw a straight line which shall make equal angles with two given intersecting straight lines and be equidistant from two given points.

4. In the figure of Prop. 1 produce AB both ways to meet the circumferences in D and E. Join CD, CE. Prove that CDE is an isosceles triangle having one angle four times each of the other angles.

5. The angle ABC of the triangle ABC is bisected by BD, which meets AC in D; and through D, DE