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

### From inside the book

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**middle points**of AB , BC , CD , DA , and EF , FG , GH , HE are joined . Prove that EFGH has all its sides equal . 12. Prove by superposition that the squares described on two equal straight lines are equal . 13. If two quadrilaterals ... Page 29

... points D , E are taken equally distant from A ; prove that the triangles ABE , ACD are equal in all respects , and also the triangles DBC , ECB . 6. Prove that the opposite angles of a rhombus are equal . 7. D and E are the

... points D , E are taken equally distant from A ; prove that the triangles ABE , ACD are equal in all respects , and also the triangles DBC , ECB . 6. Prove that the opposite angles of a rhombus are equal . 7. D and E are the

**middle points**... Page 71

... points . Similarly , a six - sided figure may be divided into four ( that is , 6 - 2 ) B triangles ; and generally a ...

... points . Similarly , a six - sided figure may be divided into four ( that is , 6 - 2 ) B triangles ; and generally a ...

**middle point**of the hypotenuse of a right- angled triangle is equidistant from the three vertices . 8. Hence ... Page 75

... point either within or without the [ m . = 14. The straight line joining the

... point either within or without the [ m . = 14. The straight line joining the

**middle points**of any two sides of a triangle is || the third side and half of it . 15. If the**middle points**of the three sides of a triangle be joined with ... Page 80

... point in AD or AD produced , and F any point in BC or BC produced ; AF , DF , BE , CE are joined . Prove △ AFD = = A BEC . 7. ABC is any triangle ; L and K are the

... point in AD or AD produced , and F any point in BC or BC produced ; AF , DF , BE , CE are joined . Prove △ AFD = = A BEC . 7. ABC is any triangle ; L and K are the

**middle points**of AB and AC ; BK and CL are drawn intersecting at G ...### 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.