5. The square described on the diagonal of a given square is double of the given square.

6. If the opposite angles of a quadrilateral figure are supplementary, a point can be found which is equidistant from the four vertices.

XLII.

1. Of all triangles having the same base and area, the perimeter of an isosceles triangle is least.

2. Show how to construct a triangle having its medians equal to three given lines.

3. If the base BC of a triangle ABC be divided into any number of equal parts at the points P, Q, R, and these points be joined to the vertex A, show that any line parallel to BC will be divided into equal parts by the lines AP, AQ, AR.

4. Show that Prop. 47 may be proved by cutting off four right-angled triangles from each of two equal squares.

5. ABC is an equilateral triangle, and AD is drawn perpendicular to BC. Prove that the square on AD is equal to three times the square on BD or CD.

6. If the sum of one pair of opposite sides of a convex quadrilateral figure is equal to that of the other two sides, a point can be found which is equidistant from the four sides.

PART VIII.

TO EUCLID I. 48.

XLIII.

1. Given four lines, no two of which are parallel. In how many points will these lines intersect, and how many diagonals can be drawn joining two points of intersection. Draw such a complete quadrilateral, and, name its sides and diagonals.

2. O is any point outside the parallelogram ABCD, and also outside the angle BAD and its opposite vertical angle. Prove that the triangle DAC will be equal to the sum of the triangles OAD, OAB.

3. Assuming the rider in XLII. 3, show how to divide a given straight line into any given number of equal parts.

4. In a right-angled triangle if a perpendicular be drawn from the right angle to the base, the square on either of the sides containing the right angle is equal to the rectangle contained by the base, and its segment adjacent to that side.

5. AB and CD are two given finite straight lines. Draw BE at right angles to AB, and equal to CD. Show how to find a point H in AB, such that the

difference of the squares on AH and HB shall be equal to the square on CD.

6. In any triangle if a perpendicular be drawn from one extremity of the base to the bisector of the vertical angle, the line joining the middle point of the base to the foot of this perpendicular is equal to half the difference of the sides of the triangle.

XLIV.

Two of its

1. ABCD is a quadrilateral figure. opposite sides AD and BC are bisected at X and Y ; and its diagonals AC and BD are bisected at Z and W. Prove that XZWY is a parallelogram whose area is equal to half the difference of the areas of the triangles ABC and ABD.

2. ABCD is a parallelogram, and O is any point within the angle BAD or its opposite vertical angle. Prove that the triangle OAC is equal to the difference of the triangles OAD, OAB.

3. Any straight line drawn from the vertex of a triangle to the base is bisected by the straight line which joins the middle points of the other sides of the triangle.

4. ABC is a right-angled isosceles triangle, having the side AB equal to BC. If BC is produced to D, E and F, making BD equal to AC, BE equal to AD, and BF equal to AE, show that the squares on BD, BE and BF are equal to twice, three times and four times the square on AB.

5. Given the base of a triangle, and the difference of the squares on the sides of the triangle. Show that

GCL

the locus of the vertex of the triangle is a straight line perpendicular to the base.

6. Assuming the last rider, No. 5, prove that the three perpendiculars from the angles of a triangle to the opposite sides are concurrent.

XLV.

1. Assuming the figure and rider XLIV. 1, if AB and DC meet in L, and ZY produced meets LC in K; prove that each of the triangles LZY and CZY is equal to one-fourth of ABC; that each of the triangles LWY and BWY is equal to one-fourth of BCD; and that the triangle LZW is equal to one-fourth of the quadrilateral figure ABCD.

2. In a triangle ABC, AD is drawn perpendicular to BC, and X, Y, Z are the middle points of the sides BC, CA, AB. Prove that each of the angles ZXY, ZDY is equal to the angle BAC.

3. If two straight lines AB, CD intersect in O, so that the triangle AOC is equal to the triangle DOB, prove that AD and CB are parallel.

4. Show how to divide a given straight line into two parts, so that the square of one part may be double of the of the other part. square

5. In the figure of Prop. 47 join FD and EK, and prove that the square on FD is equal to the square on AB together with four times the square on AC.

6. Construct a square so that one side shall lie on a given straight line and two other sides shall pass through two given points.

XLVI.

1. Assuming XLV. 1, show that the middle points of the three diagonals of a complete quadrilateral are collinear, i.e. in the same straight line.

2. The perpendiculars through the middle points of the sides of a triangle ABC meet in P, and the medians meet in M. Join PM and produce it to meet AD, the perpendicular from A to BC, in O. Prove that MO= twice PM, and that all the perpendiculars from the angles of the triangle pass through O.

3. The angles ABC and ACB are bisected by the lines BK and CK, and DKE is drawn through K parallel to BC meeting AB and AC in D and E. Prove that DE is equal to the sum of BD and CE.

f

4. Show how to divide a given straight line into two parts so that the square of one part may be equal to three times the square of the other part.

5. In the figure of Prop. 47 join FD and EK and prove that the squares on FD and EK are equal to five times the square on BC.

6. Construct a square so that two opposite sides shall pass through two given points, and its diagonals intersect at a third given point.

XLVII.

1. The quadrilateral figure, which is formed by the four straight lines bisecting the angles of any quadrilateral figure ABCD, has its opposite angles equal to two right angles.

D