...proof) any one subsequent proposition in the course of the proof of which reference is made to it. 3. Equal triangles on the same base, and on the same side of it, are between the same parallels. Prove that diameters (produced, if necessary) of any pair of parallelograms about the diameter of a...

...are equal (ax. 7). .-. A ABC = A DEF. Wherefore triangles, &c. — QED PROPOSITION XXXIX. THEOREM. Equal triangles on the same base and on the same side of it a re between the same parallels. Let the equal As ABC, DBC be on the same base BC and on the same side...

...it. 4. If a line (AB) be divided into any two segments at C, prove AB2+BC2=2AB.BC+AC3. SECTION B. 6. Equal triangles on the same base and on the same side of it lie between the same parallels. 7. On a given line construct a rectangle equal to a given rectilineal...

...half Q GC (/. 34), and A DEF is half Q EH, .-. A ABC = A DEF (Ax. 7). QED HU TYTv PROP. 39. THEOK. Equal triangles on the same base and on the same side of it are between the same parallels. Given A ABC = A DBC, on same AD base BC. To prove AD || BC. If AD is not || BC, through A draw AE ||...

...an axiom and a theorem. 8. Any two sides of a triangle are together greater than the third side. 4. Equal triangles on the same base and on the same side of it, are between the same parallels. 5. If a straight line be divided into two equal, and also into two unequal parts, the squares on the...

...is equal to three times the arc AE. 230. The straight lines which bisect the vertical angles of all triangles on the same base and on the same side of it, and having equal vertical angles, all intersect at the same point. 231. If two circles touch each other...

...to those at (7 and D, show that two sides of the quadrilateral are parallel to one another. (10) 10. Equal triangles on the same base and on the same side of it are between the same parallels. If a quadrilateral is divided into four equal triangles by its diagonals, show that it is a parallelogram....

...isosceles that is BA=BE, but BE = AC. Л the triangle ABC is isosceles. Q- b- °2. Equal triangles upon the same base and on the same side of it are between the same parallels. Prop. 39, Bk. I. 3. To a given straight line to apply a parallelogram, which shall be equal to a given...

...semicircle thus formed, and the segments of a circle greater than a semicircle. (7) ABC and DEC are two triangles on the same base and on the same side of it, the vertex of each triangle being without the other; if the vertices be joined name the quadrilateral...

...equal to L BCD, which is impossible. The third case needs no proof, because BC is not = BD. Hence two triangles on the same base and on the same side of it cannot have their conterminous sides equal. 1. On the same base and on the same side of it there can...