## The Elements of Euclid, books i. to vi., with deductions, appendices and historical notes, by J.S. Mackay. [With] Key1884 |

### From inside the book

Results 11-15 of 78

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**less**than three right angles . 9. The three exterior angles of a triangle made by producing the sides in succession , are together greater than three right angles . Prove by indirect demonstrations the following theorems : 10. The ... Page 47

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**less**than another side , the angle opposite to it must be acute . 4. ABCD is a quadrilateral whose longest side is ...**less**than AB . B = L C. I. 5 If AC be**less**than AB , then △ B Book I. ] 47 PROPOSITIONS 18 , 19 . Page 48

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**less**than C. I. 18 But it is not ; ... AC is not**less**than AB . Hence AC must be greater than AB . COR . — The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line ; and of others ... Page 50

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**less**than the sum of the other three sides . 7. Any side of a polygon is**less**than the sum of the other sides . S. The sum of the distances of any point from the three angles of a triangle is greater than the semi - perimeter of the ... Page 51

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**less**than twice the sum of the two diagonals . 12. The sum of the diagonals of a quadrilateral is**less**than the sum of the four straight lines which can be drawn to the four angles from any other point except the intersection of the ...### Other editions - View all

### Common terms and phrases

ABē ABCD ACē ADē angles equal base BC bisected bisector CDē centre chord circumscribed Const deduction diagonals diameter divided in medial divided internally draw equiangular equilateral triangle equimultiples Euclid's exterior angles Find the locus given circle given point given straight line greater Hence hypotenuse inscribed intersection isosceles triangle less Let ABC lines is equal magnitudes medial section median meet middle points opposite sides orthocentre parallel parallelogram perpendicular polygon produced PROPOSITION 13 Prove the proposition quadrilateral radical axis radii radius ratio rectangle contained rectilineal figure regular pentagon required to prove rhombus right angle right-angled triangle square on half straight line drawn straight line joining tangent THEOREM unequal segments vertex vertical angle Нур

### Popular passages

Page 147 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

Page 276 - IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. NB This is usually cited by the words

Page 331 - If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another...

Page 17 - From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, whereof AB is the greater.

Page 112 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Page 87 - Guido, with a burnt stick in his hand, demonstrating on the smooth paving-stones of the path, that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

Page 254 - If there be four magnitudes, and if any equimultiples whatsoever be taken of the first and third, and any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the first is greater than the multiple of the second, equal to it or less, the multiple of the third is also greater than the multiple of the fourth, equal to it or less ; then, the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth.

Page 138 - RULE. from half the sum of the three sides, subtract each side separately; multiply the half sum and the three remainders together, and the square root of the product will be the area required.

Page 304 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Page 44 - America, but know that we are alive, that two and two make four, and that the sum of any two sides of a triangle is greater than the third side.