 | John Radford Young - Euclid's Elements - 1827 - 246 pages
...proportional lines are proportional, ABC : DEF : : BC2 : EF2 : : AB2 : DE1 : : AC2 : DF2. Cor. Since triangles of the same altitude are to each other as their bases, it follows (Prop. XIV.) that in a right angled triangle the squares of the sides are to each other... | |
 | Timothy Walker - Geometry - 1829 - 138 pages
...half of B N. But GB is the altitude of AC B, and BP is the altitude of AM B. Accordingly, since „ triangles of the same base are to each other as their altitudes, and since GB is greater than BP, the triangle AC B is greater than AMB, which was to be demonstrated. 120.... | |
 | Dugald Stewart - 1829 - 482 pages
...circumference on the same base, we ascertain a relation between two quantities. When we demonstrate, that triangles of the same altitude are to each other as their bases, we ascertain a connexion between two relations. It is obvious, how much the mathematical sciences must... | |
 | John Playfair - Euclid's Elements - 1835 - 336 pages
...proportional to M, N ; draw the line AD, and the triangle ABC will be divided as required. For, since the triangles of the same altitude are to each other as their bases, we have ABD : ADC : : BD : DC : : B. D 0 M:N. SCHOLIUM. A triangle may evidently be divided into any... | |
 | Adrien Marie Legendre - Geometry - 1836 - 394 pages
...Sch.) ; therefore, AB x BE A. B in equal to the area of the parallelogram ABCD. Cor. Parallelograms of the same base are to each other as their altitudes ; and parallelograms of the same altitude are to each other as their bases : for, let B be the common base,... | |
 | Adrien Marie Legendre - Geometry - 1837 - 376 pages
...therefore the solidity of a cylinder is equal to the product of its base by its altitude. Cor. 1. Cylinders of the same altitude are to each other as their bases ; and cylinders of the same base are to each other as their altitudes. Cor. 2. Similar cylinders arc to each... | |
 | John Playfair - Euclid's Elements - 1837 - 332 pages
...proportional to M, N ; draw the line AD, and the triangle ABC will be divided as required. For, since the triangles of the same altitude are to each other as their bases, we have ABD : ADC : : BD : DC : : M: N. SCHOLIUM. PROP. Q. PROS. To divide a triangle intn two parts... | |
 | Benjamin Peirce - Geometry - 1837 - 216 pages
...its altitude. 252. Corollary. Triangles of the same base are to each other as their altitudes, and triangles of the same altitude are to each other as their bases. 253. Theorem. The area of a trapezoid is half the product of its altitude by the sum of its parallel... | |
 | Adrien Marie Legendre - Geometry - 1838 - 382 pages
...parallelogram is equal to BC x AD (Prop. V.) ; hence that of the triangle must be iBC x AD, or BC x |AD. Cor. Two triangles of the same altitude are to each...other, as the products of their bases and altitudes. BOOK IV. 75 PROPOSITION VII. THEOREM. The area of a trapezoid is equal to its altitude multiplied by... | |
 | Adrien Marie Legendre - Geometry - 1839 - 372 pages
...parallelogram is equal to BC x AD (Prop. V.) ; hence that of the triangle must be iBC x AD, or BC x Cor. Two triangles of the same altitude are to each other as iheir bases, and two triangles of the same base are to each other as their altitudes. And triangles... | |
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Two triangles of the same altitude are to each other as their bases, and two triangles of the same base are to each other as their altitudes. And triangles generally, are to each other, as the products of their bases and altitudes. 
