...Therefore the triangle BDE is equal to the triangle CDE (V. 9); and they are on the same base DE. But equal triangles on the same base and on the same side of it are between the same parallels (L 39). Therefore DE is parallel to BC. Wherefore, if a straight line, &c. QED PROPOSITION III.—...

...DAB are equivalent, therefore EAD and DAB would be equivalent, which is absurd. COR. Hence equivalent triangles on the same base, and on the same side of it, must be between the same parallels. ELEMENTARY GEOMETRY. THEOREM 25. If a straight line is divided...

...on the same side of that line are between the same parallels. Corollary 2. — Equal triangles upon the same base and on the same side of it are between the same parallels. Corollary 3. — Equal parallelograms upon the same base and upon the same side of it are between the...

...unalterably to a certain shape, from the fact that we have now two triangles, and that there cannot be two triangles on the same base and on the same side of it, and having their other sides respectively equal. The tightness of the fastenings and the caulking of...

...in every respect ? 2. Equal triangles, standing on equal bases, situated ?T? the same straight line, and on the same side of it, are between the same parallels ? 3. If a right line be divided, the sum of the squares of the \i hole line and of one segment is equal...

...and therefore are also equal (by Ax. 7). PROPOSITION XXXIX. THEOREM. Equal triangles (BAC and BDC) on the same base, and on the same side of it, are between the same parallels. For if AD be not parallel to BC, draw through the point A the right line AF parallel to BC, cutting the side...

...the point of intersection of the diagonals, are equivalent. 3. Equivalent triangles or parallelograms on the same base and on the same side of it are between the same parallels. 4. If through any point in the diagonal of a parallelogram lines are drawn parallel to the sides, the...

...parallelograms are equal, by Prop. 36. Therefore the triangles are equal, by Ax. 1. Proposition 39. — Theorem. Equal triangles on the same base and on the same side of it are between the same parallels. If ABC = DBC, AD is parallel to BC. If not, let DE be parallel to BC, and let it cut AC, or AC produced,...

...therefore 2. The triangle BDE is equal to the triangle CDE (V. 9) ; and they are on the same base DE ; but equal triangles on the same base and on the same side of it, are between the same parallels (I. 39) ; therefore 3. DE is parallel to BC. WTierefore, if a straight line, &c. QED PROPOSITION 3.—...

...the triangle DEF. Therefore, triangles, <fec. QED Proposition 89. — Theorem. Equal triangles upon the same base, and on the same side of it, are between the same parallels. Let the equal triangles ABC, DBC be upon the same base . BC, and on the same side of it ; They shall...