......... (I) and since the altitude aD is the same f»r the three triangles ODa, OMa, OAa ; and that triangles of the same altitude are to one another as their bases, it follows from the above proportion (1), that ODa : OMa :: OMa : OAa Consequently, the numerical measure...

...simultaneous division. Wherefore, if two triangular pyramids, &e. QED PROP. V. THEOREM. Triangular pyramids of the same altitude, are to one another as their bases. Let the pyramids AB С G and DEFH be of the same altitudes. The base ABС is to the base DEF, as the pyramid...

...the solid AB to the solid CD. COROLLARY. From this it is manifest, that prisms upon triangular bases, of the same altitude, are to one another as their bases. Let the prisms, the bases of which are the triangles AEM, CFG, and NBO, PDQ the triangles opposite to them,...

...COROLLARY. — Two triangles of the same base are to one another в i> as their altitudes ; and two triangles, of the same altitude, are to one another as their bases. THEOREM 45. 115. The area of every trapezoid, ABCD, is equal to the product of half the sum of its...

...Show how to describe a circle touching one side of a triangle and the other two sides produced, 2. Triangles of the same altitude are to one another as their bases. In what ratio are the areas of the three triangles formed by joining the centre of the inscribed circle...

...double of the third angle. How many degrees will there be in the several angles of such a triangle 1 2. Triangles of the same altitude are to one another as their bases. 3. If AC be a diameter of. the parallelogram ABCD, show that the perpendicular distances of the points...

...the solid AB to the solid CD. COBOLIARY. From this it is manifest, that prisms upon triangular bases, of the same altitude, are to one another as their bases. Let the prisms, the bases of which are the triangles AEM, CFG, and NBO, PDQ the triangles opposite to them,...

...equal to one another. 4. To inscribe an equilateral and equiangular pentagon in a given circle. 5. Triangles of the same altitude are to one another as their bases. 6. If a, b, c, be the sides of the triangle ABC, prove that a2=Z-2 + c2-26ecos A. If the sides of a...

...semicircle. 8. When, according to Euclid, are four magnitudes proportional 1 show geometrically that triangles of the same altitude are to one another as their bases. Point out how it may be shown in a similar manner that triangles with equal bases are to another as...

...rectangles and parallelograms of the same altitude are to one another as their bases ; and then deduced that triangles of the same altitude are to one another as their bases : — does it appear that any advantages would have arisen in this mode of treating the subject ? 67....