...equal to one another, from which we learn another geometrical fact, namely, that all angles standing on the same base and on the same side of it, and having their tops or vertices in the circumference of a circle, are equal to one another. In Case 4, where...

...quadrilateral rectilineal figure inscribed in the same circle. PROP. XIII. 73. Of all triangles standing upon the same base, and on the same side of it, and having equal vertical angles, that which is isosceles is the greatest. Let ACB be an isosceles triangle, and APB any other triangle...

...parallelograms formed of which the given triangles as halves they are .-. equal, PROP. 39, THEOR. Equal triangles on the same base and on the same side of it are between the same parallels, For if they are not, draw through the vertex of one of them a line...

...are =, the vertices lie in the same segment standing on the given base. 362. Of all ^\* standing upon the same Base, and on the same side of it, and having = vertical L *, that which is Isosceles has the greatest perimeter. Use Propp. XXI., Book III. ; XX.,...

...a point in which the circumferences meet, the circles must touch one another in that point. IF two triangles on the same base, and on the same side of it, have equal vertical angles, the vertex of each is in the circumference of the circle described about...

...therefore be also greater than Z. BDC, of which however it is but a part ; which is absurd. Therefore, two triangles on the same base, and on the same side of it, cannot have their conterminous sides equal at both extremities of the base, when the vertex of the...

...therefore* the triangle BDE is* 9. 5. equal to the triangle CDE: 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 ;* * 39. i therefore DE is parallel to BC. Wherefore, if a straight...

...to different extremities of the other; shew that two triangles are formed which are equiangular. 31. The lines, which bisect the vertical angles of all triangles on the same base and with the same vertical angle, all intersect in one point. 32. A number of triangles are described upon...

...circumferences meet, the circles must touch one another in that point. PROP. III. THEOR. — If two triangles on the same base, and on the same side of it, have equal vertical angles, the vertex of each is in the circumference of the circle described about...

...PLANE GEOMETRY. other lino passing through A, except AD, which therefore is || to BF. QED Cor. 1. Equal triangles on the same base and on the same side of it, are between the same parallels. Cor. 2. In the same manner it might be shown that equal triangles between...