2. ACB and ADB are two triangles on the same base AB and on the same side of it, and AC is equal to BD, and AD to BC. If AD and BC intersect in P., prove that the triangle APB is isosceles.

3. Find a point within a given triangle equidistant from the three sides.

4. Straight lines which make a given acute or obtuse angle with a given straight line form two sets of parallel lines.

5. If one angle of a triangle is equal to the sum of the other two angles, the triangle is right angled.

6. Show how to trisect a right angle.

PART IV.

TO EUCLID I. 34.

XIX.

1. AB, AC are two straight lines. Through the given point X draw a straight line meeting AB and AC in D and E and making AD equal to AE.

2. Construct a triangle having given the base, the altitude and the length of the median which bisects the base.

3. In a right-angled triangle if a perpendicular be drawn from the right angle to the hypothenuse, the two triangles thus formed are equiangular to one another.

4. Every right-angled triangle can be divided into two isosceles triangles by a straight line drawn from the right angle to the hypothenuse, and this line is equal to half the hypothenuse.

5. What is the magnitude of each of the angles of a regular pentagon ?

6. If the diagonals of a parallelogram are equal all its angles are right angles.

XX.

1. If in Prop. 33 the lines were joined, but not towards the same parts, state and prove what difference there would be in the conclusion.

2. AB and C are two given straight lines. At the point B the angle ABD is made equal to half a right angle. Find a point P in BD, and a point Q in AB, so that AQP shall be a right-angled triangle having its hypothenuse equal to C, and the sum of its sides equal to AB.

3. A is the vertex of an isosceles triangle. Produce BA to D, making AD equal to AB; and join CD. Prove that BCD is a right angle.

4. A number of right-angled triangles have a common right angle and equal hypothenuses. Show that the middle points of the hypothenuses all lie on the circumference of the same circle.

5. What is the magnitude of each of the angles of a regular hexagon?

6. Two straight lines drawn from the extremity of the base of any triangle cannot bisect each other.

XXI.

1. In the figure of Prop. 16 if the angle ABC is bisected by the line BX meeting AC in X, prove that the median BE falls within the angle ABX so long as AB is greater than BC.

2. If ABC is a triangle having the angles at A and B equal to half two given angles P and Q, and if at the point C the angles ACD, BCE be described equal

respectively to half P and Q, the lines CD, CE meeting the base AB within the triangle, then CDE will be a triangle having its perimeter equal to AB, and the angles at the base equal to P and Q.

3. Draw a straight line at right angles to a given finite straight line from one of its extremities without producing the given straight line.

4. Prove indirectly that if the bisectors of two angles of a triangle are equal the two angles are equal.

5. What is the magnitude of each of the exterior angles of a regular octagon ?

6. The straight lines which bisect two opposite angles of a parallelogram are either coincident or parallel.

XXII.

1. In the figure of Prop. 16 if the angle ABC be bisected by BX, and BP be drawn perpendicular to AC, prove that the bisector BX is intermediate in position and magnitude to the median BE and the perpendicular BP so long as AB and BC are unequal.

2. If ABC is a triangle having the angle at B equal to half a given angle P, and the side AC equal to a given line K, show how to describe on the base AB a triangle having the difference of the base angles equal to P and the difference of the sides equal to K.

3. A parallelogram is bisected by any straight line which passes through the middle point of one of its diagonals.

4. If one angle of a parallelogram is a right angle all its angles are right angles.

5. If the opposite sides of a quadrilateral figure are equal it is a parallelogram.

6. The straight lines which bisect two adjacent angles of a parallelogram intersect at right angles.

XXIII.

1. In any triangle the angle contained by the bisector of the vertical angle and the perpendicular from the vertex to the base is equal to half the difference of the angles at the base of the triangle.

2. ABCD is a quadrilateral figure having AB parallel to CD. Prove that its area is equal to the area of a parallelogram formed by drawing through M the middle point of BC a straight line parallel to AD.

3. From the extremities of the base of the isosceles triangle ABC, BP and CQ are drawn perpendicular to the equal sides AC and AB. Prove that each of the angles PBC and QCB is equal to half the angle BAC.

4. The diagonals of a parallelogram bisect each other. 5. If the opposite angles of a quadrilateral figure are equal it is a parallelogram.

6. The lines which bisect the angles of any parallelogram form a right-angled parallelogram, whose diameters are parallel to the sides of the former parallelogram.

XXIV.

1. On a given base construct a triangle, having one angle equal to a given angle A, and the side opposite this angle equal to a given straight line B. When will