Ex. 82. Perform the above construction. Calculate what should be the magnitudes of the angles of the triangle, and verify that your figure agrees with your calculation. (To save time, it will be best to divide AB in the required manner arithmetically, i.e. by measuring off the right length.)

Ex. 83. Show that, in fig. 141, BD is the side of a regular decagon inscribed in the circle.

Ex. 84. Show that, if ACD is drawn, BD will be a tangent to that circle. Ex. 85. Prove that AC and CD are sides of a regular pentagon inscribed in ACD.

Ex. 86. Let DC be produced to meet the circle of fig. 141 in E. Prove that BE is the side of a regular 5-gon inscribed in the circle.

Ex. 87. Prove that AE= EC. (See Ex. 86.)

Ex. 88. Prove that AE is || to BD. (See Ex. 86.)

Ex. 89. Prove that as AED, CAD are similar. (See Ex. 86.)

Ex. 90. Prove that DE is divided in extreme and mean ratio at C. (See Ex. 86.)

Ex. 91. Prove that, if ABD is drawn, BD is the side of a regular pentagon inscribed in the .

Ex. 92. Let the bisectors of 48 B, D meet ABD in F, G. Prove that AGBDF is a regular pentagon.

To construct a regular pentagon

Construction Construct an isosceles ▲ ABC with each of its base angles twice the vertical angle. Draw the circumscribing of AABC.

Then BC is a side of a regular 5-gon inscribed in O ABC.

Proof Since Abc = acb=2▲ BAC,

... 4 BAC=1 of 2 rt. s = 36°.

Fig. 142.

... BC subtends 36° at circumference and 72° at centre.
... BC is a side of a regular 5-gon inscribed in the O.
The pentagon may now be completed.

SOLID.

Ex. 93. Three points X, Y, Z are taken on each of the edges OA, OB, OC of a cube. Show that the angles of the triangle XYZ are all acute.

Ex. 94. Prove that the square of the distance between two points is equal to the sum of the squares of its projections on three straight lines at right angles to one another.

S. H. T. G.

10

Ex. 95. The sum of the plane angles at the vertex of any pyramid is less than four right angles.

Ex. 96. If any solid angle is contained by three plane angles, the sum of any two of the plane angles is greater than the third angle.

Ex. 97. The sum of the angles of a skew quadrilateral is less than four right angles.

Ex. 98. If each edge of a tetrahedron is equal to the opposite edge, show that the sum of the plane angles at each corner is two right angles.

Ex. 99. The four diagonals of a parallelepiped* are concurrent and each is bisected at this point.

Ex. 100. The sum of the squares on the four diagonals of a parallelepiped is equal to the sum of the squares on the twelve edges.

THE TETRAHEDRON.

Ex. 101. Show that the figure obtained by joining the mid-points of adjacent sides of a skew quadrilateral is a parallelogram.

Deduce that the lines joining the mid-points of opposite edges of a tetrahedron intersect in a point which bisects each line.

Ex. 102. Use the circumscribed parallelepiped to prove that joins of midpoints of opposite edges of a tetrahedron co-intersect and bisect one another.

Ex. 103. Prove that any plane parallel to two opposite edges of a tetrahedron cuts the faces in a parallelogram, whose angles are the same for different planes. Also that, if two edges are at right angles, the section may be a square.

Ex. 104. If two pairs of opposite edges of a tetrahedron are at right angles, the third pair also are at right angles.

Ex. 105. If each edge of a tetrahedron is equal to the opposite edge, the straight line joining the middle points of any two opposite edges is at right angles to each of those edges.

Ex. 106. If two of the joins of mid-points of opposite edges of a tetrahedron are at right angles, the remaining edges are equal.

Ex. 107. Show that if the sum of the squares of two opposite edges of a tetrahedron is equal to the sum of the squares on another pair of opposite edges, the two remaining opposite edges are at right angles to one another.

* A parallelepiped is a figure bounded by three pairs of parallel planes. A brick is a rectangular parallelepiped.

+ Many properties of the tetrahedron can be proved by means of the circumscribed parallelepiped constructed as follows:

Through each pair of opposite edges of a tetrahedron construct a pair of parallel planes; this gives a parallelepiped circum

Fig. 143.

scribed to the tetrahedron, the edges of the latter being diagonals of the faces of the former.

Ex. 108. Prove that the line joining the mid-points of two opposite edges of a regular tetrahedron is perpendicular to each edge (i.e. is the shortest distance between them) and is one-half of the diagonal of the square on an edge.

Ex. 109. The sum of the squares on the edges of any tetrahedron is four times the sum of the squares on the straight lines joining the middle points of opposite edges. [See Ex. 102.]

Ex. 110. The straight lines which join the vertices of a tetrahedron to the centroids of the opposite faces meet in a point which is a point of quadrisection of each line. [This point is the centroid of the tetrahedron.]

Ex. 111. The sum of the squares on the edges of any tetrahedron is four times the sum of the squares on the lines joining the vertices to the centroid of the tetrahedron.

Ex. 112. In a regular tetrahedron prove that three times the square on the perpendicular from any vertex to the opposite face is equal to twice the square on an edge.

Ex. 118. Prove that four points determine a sphere uniquely; that is, a sphere can be circumscribed about a tetrahedron +.

Ex. 114. Prove that eight spheres (in general) exist, each of which touches the four planes forming the faces of a tetrahedron.

Ex. 115. If a sphere exists which touches all the edges of a tetrahedron within their unproduced portion, the three sums of opposite edges are equal. What is the intersection of this sphere with a face?

Ex. 116. In the case of a regular tetrahedron, compare the radii of the circumscribed sphere, the inscribed sphere, and the sphere touching the six edges at their middle points.

*See definition, p. 136, Ex. 4.

+ See Practical Geometry, p. 239, Ex. 57.

REVISION PAPERS

Paper 1.

1. D is the middle point of the side BC of a triangle ABC. If DA is equal to half BC, prove that the angle BAC is equal to the sum of the angles B and C. 2. The internal bisectors of two angles of a triangle can never be at right angles to one another.

3. ABCD is a parallelogram and BE is the bisector of the angle ABC. DE, drawn perpendicular to BE, is produced (backwards if necessary) to meet BC produced in P. Prove that the triangle PDC is isosceles.

4. The extremities of a pair of opposite sides of a rectangle are joined to the mid-points of the opposite sides. Prove that the parallelogram formed by the intersections of these lines is of the area of the rectangle.

5. Chords AP, BQ are drawn perpendicular to a chord AB at its extremities. Prove that AP=BQ.

6. If straight lines are drawn from a point perpendicular to the arms of an angle, the angle between those straight lines is equal or supplementary to the given angle.

7. Find the locus of the point of intersection of tangents to a circle which meet at a constant angle.

8. A line PQ is drawn through the point of contact of two circles which touch one another, cutting these circles in P and Q. Prove that the tangents at P and Q are parallel.

9. The bisectors of the base angles B, C of an equilateral triangle ABC meet at D. Through D lines are drawn parallel to the sides AB, AC of the triangle, meeting the base in E and F. Prove that E and F trisect BC.

10. Two circles PAB, QAB cut in A and B, and O, the centre of the latter, lies on the former. Show that, if R is any point on the common chord AB, and if OR produced meets the circle PAB in P, then OR. OP=OQ2.

Paper 2.

1. The bisectors of the interior angles at B and C of the triangle ABC meet the bisector of the exterior angle at A in the points P, Q respectively, and BP, CQ meet at O. Prove that OPQ=ACO and ▲ OQP = LABO.

2. The quadrilateral ABCD has the angles at A and B supplementary and the sides AD, BC equal. Prove that it must be a parallelogram.

If the angles at A and B are supplementary, but the sides AB, CD equal, prove that the figure is either a parallelogram or a trapezium which has the angles at A and D equal and likewise those at B and C equal.

3. What is the locus of the vertices of triangles on a given base and of given area? Quote the theorem which justifies your statement.

A triangular field ABC has to be divided into four parts which are to be equal to one another in area. Draw any triangle to represent the field and show how to divide it so that the given conditions may be satisfied. Give a proof.

4. HVQ is a triangle right-angled at V, HVT is a triangle on the opposite side of HV having ▲ THV a right angle; prove that the squares on HT, HQ are together equal to the squares on TV, VQ.

If the triangle HVT is turned about HV till its plane is at right angles to the plane HVQ, prove that the square on TQ is then equal to the sum of the squares on HT, HQ.

5. ABCD is a quadrilateral inscribed in a circle whose centre is O, and E, F are the middle points of BC, CD. Prove that EOF is either equal or supplementary to BAD.

6. ABC is a right-angled triangle in which C90°. A square is described on AB so as to be on the opposite side of AB from C. The diagonals of the square intersect in D. Prove that CD bisects the angle C.

7. APQ is a triangle obtuse-angled at Q. At P and Q, on the side away from A, construct angles QPR, PQR equal to A. Draw PO and QO at right angles to PR, QR on the same side as A and meeting at O. Prove that O is equidistant from A, P, Q.

8. Construct a triangle ABC in which a=3", c=4′′, and B=29°. Draw its inscribed circle, and also the escribed circle which touches AC between A and C. Measure the radius of each circle, and show theoretically that the line joining their centres must pass through B.

9. In the given figure ABCD, AXYZ are two squares. Prove (i) that ▲s AXB, AZ D`are congruent; (ii) that As AXB, AYC are similar.

10. ABC is a triangle inscribed in a circle and a line PQ is drawn parallel to the tangent at A cutting AB and AC at D and E. Prove B, D, E, C concyclic.

Paper 3.

D

B

X

1. In the equal sides AB, AC of an isosceles triangle ABC, show how to find points X, Y, such that BX=XY=YC. Give a proof.

2. ABCD is a parallelogram; E, F, G, H are points in AB, BC, CD, DA respectively. Being given AE=CG and BF=DH, prove that EFGH is a parallelogram.

3. A four-sided field is to be divided into two parts of equal area; prove the accuracy of the following construction. Draw a quadrilateral ABCD to represent the field; draw the diagonal AC; find E, the mid-point of AC; join BE, DE; then the areas ABED and CBED are equal.

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