2. Euclid I. 10.

In the figure of Euclid I., Prop. 1, if the two points in which the circles meet be joined, the given finite straight line will be bisected.

3. Euclid I. 36.

4. Show that if a quadrilateral be bisected by both its diagonals it is a parallelogram.

5. Euclid I. 48.

6. Euclid II. 6.

7. Prove that if ABC be a triangle, obtuse-angled at B, and D be the foot of the perpendicular from C on AB produced, the square on AC exceeds the squares on AB, BC, by twice the rectangle AB, BD.

(Questions 8, 9, 10, 11 and 12 were set on Books III., IV. and VI.)

VIII.

Cambridge Local Examinations, December 1890.
Junior Students.

Elementary Euclid. Books I., II.

1. Define a plane superficies, a plane rectilineal angle, and a right-angled triangle.

Give Euclid's Axiom relating to right angles. 2. Euclid I. 5.

If on a common base and on opposite sides of it be described two isosceles triangles, the straight line joining their vertices will cut the base at right angles. 3. Euclid I. 34.

The diagonals AC, BD of a quadrilateral ABCD intersect in O, and the parallelograms OAEB, OBFC, OCGD, ODHA are completed: prove that EFGH will be a parallelogram, and will be double of the quadrilateral ABCD.

4. Euclid I. 37.

Through A, B, C are drawn three parallel straight lines to meet the opposite sides of the triangle ABC (produced if necessary) in A', B', C': prove that the triangle A'B'C' will be double the triangle ABC.

5. Euclid I. 48.

6. Euclid II. 5.

7. Euclid II. 11.

Produce a given straight line to a point, such that the rectangle contained by the whole line thus produced and the part produced may be equal to the square on the given straight line.

IX.

Cambridge Local Examinations, December 1890.
Senior Students.

1. Euclid I. 32.

ABC is any acute angle, AB is bisected in D, and at K in BC the angle DKB is made equal to the angle DBK; if AK be drawn, prove that it is perpendicular to BC.

2. Euclid I. 34.

ABCD is a parallelogram, BOD one of its diagonals, and EOG, FOH are drawn parallel to BC, CD respectively, so that E, F, G, H lie, correspondingly, on the sides AB, BC, CD, DA. If DF, BH be drawn intersecting EG in K, L respectively, prove that OK is equal to OL.

3. Euclid II. 14.

A straight line AB is produced both ways to C and D, so that BD is twice AC: show how to find the points C and D when the rectangle CA, AD is equal to the square on AB.

(Questions 4, 5, 6 and 7 were set on Books III., IV., VI. and XI.)

X.

Oxford and Cambridge School Examinations, 1890. For Commercial Certificates.

1. If two triangles have three sides of the one equal to three sides of the other each to each, the triangles are equal to one another in every respect.

Prove that the diagonals of a rhombus bisect one another, and cut one another at right angles.

2. Euclid I. 22.

Show how the construction would fail if two of the straight lines were together not greater than the third.

3. Euclid I. 32.

Show that each angle of a regular polygon with fifteen sides is twenty-six fifteenths of a right angle. 4. Euclid I. 46.

5. Euclid II. 11.

7. Euclid II. 13.

(Questions 6 and 8 were set on Book III.)

XI.

Oxford and Cambridge School Examinations, 1890. For Lower Certificates.

1. Define a parallelogram, a plane, a circle.

Euclid I. 7.

ACB, ADB are two triangles on the same side of AB, such that AC is equal to BD and AD is equal to BC, and AD and BC intersect in R; prove that the triangles ARC and BRD are equal in all respects.

2. Euclid I. 16.

3. Euclid I. 33.

If two sides of a quadrilateral be parallel and unequal and the other two sides be equal, the diagonals are equal.

4. Euclid I. 43.

5. Euclid II. 4.

6. Euclid II. 14.

Divide a given line into two parts so that the rectangle contained by the parts shall be equal to a given square. When is this impossible?

(Questions 7, 8, 9 and 10 were set on Books III., IV. and VI.)

XII.

Oxford and Cambridge School Examinations, 1890.
For Higher Certificates.

1. Euclid I. 10.

Prove that the two straight lines which join the middle points of the sides of an isosceles triangle to the middle point of the base are equal to one another.

2. Euclid I. 29.

The side BC of a triangle ABC is produced to D. Show that the straight lines which bisect the angles BAC, ACD cannot be parallel.

3. Euclid I. 47.

Prove that if the diagonals of a quadrilateral are at right angles the squares on two opposite sides are together equal to the squares on the other two sides. 4. Euclid II. 11.

Prove that if a straight line be divided as above the rectangle contained by the two parts is equal to the difference of the squares on the two parts.

5. Define a plane superficies, a circle, a rectilineal figure. Show that the distance between the centres of two circles whose circumferences cut one another is less than the sum, and greater than the difference of their radii. Prove that a quadrilateral cannot have all its angles obtuse.

6. Euclid I. 24.

7. Euclid I. 36.

8. Euclid II. 5.

XIII.

Admission to the R. M. Academy, Woolwich, June 1890.

1. Euclid I. 12.

2. Euclid I. 32.

Draw a straight line DE parallel to the base BC of a triangle to cut the side AB in D and AC in E, so that DE may be equal to BD and CE together.

3. D is a point in the side AB of a triangle. Find a point E in the side BC such that the triangles EAD, CAE may be equal.

4. Euclid I. 47.

Make a square which is three times the square on a given straight line.

5. Euclid II. 11.

6. Give a geometrical proof of the algebraic formula:(a+b)2+(a - b)2=2(a2+b2).

(Questions 7, 8, 9, 10, 11 and 12 were set on Books III., IV. and VI.)

XIV.

Admission to the R. M. Academy, Woolwich,
November 1890.

1. Euclid I. 5.

2. Euclid I. 27.