3. ABC is an isosceles triangle. From the equal sides AB and AC cut off BD equal to CE, and join CD and BE. Prove that CD=BE. 4. Two isosceles triangles ABC and DBC are on the same base BC. Prove that the angle DBA is equal to the angle DCA. 5. Draw two isosceles triangles ABC, ACD, such that AB AC=AD, and let them have AB and AD in one straight line. 6. In the figure of Question 5, prove that the angle BCD is equal to the sum of the angles ABC and ADC. PART II. TO EUCLID I. 26. VII. 1. Upon a given finite straight line describe an isosceles triangle having each of its equal sides double of the base. 2. In the figure of Prop. 5, join FG, and prove that the angle BGF is equal to the angle CFG. 3. The straight line which bisects the vertical angle of an isosceles triangle also bisects the base, and is perpendicular to it. 4. Let the straight line AB make with the straight line CD the angles ABC and ABD, and let these angles be bisected by the straight lines BE and BF. Prove that EBF is a right angle. 5. ABC is any triangle, and the angle BAC is bisected by the straight line AX which meets BC in X. Prove that BA is greater than BX, and CA is greater than CX. 6. ABC is any triangle. Prove that the difference between any two sides, BA and AC, is less than the third side BC. VIII. 1. AB iş a given straight line, and C and D are two points outside the line AB. Find a point X in AB, such that CX=DX. 2. In the figure of Prop. 5, let FC and BG meet in H, and join AH. Prove that AH bisects the angle BAC. 3. If two isosceles triangles are on the same base, prove that the straight line, produced if necessary, which joins their vertices will bisect their common base, and be perpendicular to it. 4. Two straight lines AB and CD cut each other in the point O, and the four angles at the point O are bisected by the lines OE, OF, OG and OH. Prove that OE and OG, and that OF and OH are in the same straight lines; and that EOG and FOH cut each other at right angles. 5. AB is a straight line, and C a point without it. Draw CD perpendicular to AB, and prove that CD is less than any other line, such as CE, drawn from C to AB. 6. Take any point O inside the triangle ABC, and join OA, OB, OC. Prove that OA, OB and OC are together greater than half the sum of AB, BC and CA. IX. 1. Show how to draw a straight line any point in which is equidistant from two given points A and B. 2. ABCD is a quadrilateral figure, and its diagonals AC and BD bisect each other at right angles. Prove that ABCD is a rhombus. B 3. Show how to divide a given rectilineal angle into two parts so that one part is one-seventh of the other part. 4. A and B are two points in the same side of the line CD. Draw AP perpendicular to CD and produce it to E, making PE equal to AP. Join BE, cutting CD in X; and join AX. Prove that AX and BX make equal angles with CD. 5. ABCD is a quadrilateral figure of which AD is the longest side and BC the shortest. Prove that the angle ABC is greater than the angle ADC, and the angle BCD greater than the angle BAD. 6. In Prop. 16 prove that the two sides AB, BC are together greater than twice the median BE, which bisects the remaining side AC. X. 1. Show how to find a point equidistant from three given points, which are not in the same straight line. 2. In an isosceles triangle two of the medians are equal. 3. From two given points on opposite sides of a given straight line show how to draw two straight lines which shall meet in the given straight line and make equal angles with it. 4. In any triangle the sum of the medians is less than the perimeter of the triangle. 5. O is any point within the triangle ABC. Prove that OA, OB, OC are together less than AB, BC, and CA together. 6. ABC is any triangle. Through P, the middle point of AB, draw any straight line QPR meeting CB in Q and CA produced in R. From PR cut off PN equal to PQ and join AN. Prove that the triangle APN is equal to the triangle PQB, and that the triangle ABC is less than the triangle QRC. XI. 1. Show how to find the centre of a circle which shall pass through two given points and have its radius equal to a given line. 2. The three medians of an equilateral triangle are equal. 3. Given two points A and B on the same side of a line CD. Find a point P in CD, such that the sum of AP and BP is a minimum. 4. X, Y, Z are the middle points of the sides BC, CA, AB of the triangle ABC; and YO and ZO are drawn at right angles to CA and AB. Join OX and prove that OX is perpendicular to BC. 5. If a line be divided into any two unequal parts, the distance of the point of section from the middle of the line is equal to half the difference of the two parts. 6. If two right-angled triangles have their hypothenuses equal, and one side of the one equal to one side of the other, the two triangles shall be equal in all respects. XII. 1. Find a point which is equidistant from four fixed points. When is this impossible? |