# Euclid's Elements of Geometry, Books I. II. III. IV

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1862 edition. Excerpt: ...to BC produced. Sequence.--The square on AB shall be greater than the squares on AC and CB, by twice the rectangle BC-CD. Demonstration.--1. Because the straight line BD is divided into two parts in the point C. 2. The square on BD is equal to the squares on BC and CD, and twice the rectangle BC-CD. (II. 4.) 3. To each of these equals add the square on DA 4. Therefore the squares on BD and DA are equal to the squares on BC, CD, DA, and twice the rectangle BC-CD. 5. But the square on BA is equal to the squares on BD and DA, because the angle at D is a right angle; (I. 47.) 6. And the square on CA is equal to the squares on CD and DA; (1.47.) 7. Therefore the square on BA is equal to the squares on BC and CA, and twice the rectangle BC-CD; that is, the square on BA is greater than the squares on BC and CA by twice the rectangle BC-CD. Conclusion.--Therefore, in obtuse-angled triangles, &c. Q.B.D, PROPOSITION 13.--THEOREM. In every triangle, the square on the side subtending an acute angle, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle. (References--Prop. I. 12, 16, 47; II. 3, 7, 12.) Hypothesis.--Let ABC be any triangle, and the angle at B an acute angle; and on BC, one of the sides containing it, let fall the perpendicular AD from the opposite angle. (1.12.) Sequence.--The square on AC, opposite to the angle B, shall be less than the squares on CB and BA, by twice the rectangle CB-BD. Case I.--First, let AD fall within the triangle ABC. Demonstration.--1. Because the straight line CB is divided into two parts in the point D, 2. The squares on CB and BD...show more

## Product details

• Paperback | 40 pages
• 189 x 246 x 2mm | 91g
• Miami Fl, United States
• English
• black & white illustrations
• 1236603508
• 9781236603500