...together with twice the square on BC. 4. Show that the internal bisectors of the vertical angles of all triangles, on the same base and on the same side of it, which have equal vertical angles, pass through one fixed point, and that the external bisectors pass...

...altitudes. Theor. N. Equal triangles on the same base, or on equal bases in the same straight line, and on the same side of it, are between the same parallels. Prop. 88. A trapezoid is equal to a rectangle whose base is half the sum of the two parallel sides,...

...solutions will be accepted. Mr. Ross, Senior Inspector. Mr. KELLY, District Inspector. SECTION A. 1. If two triangles on the same base and on the same side of it have one pair of conterminous sides equal to one another, the other pair of conterminous sides must...

...exterior vertical angle of an isosceles triangle be bisected the bisector is parallel to the base. 3 (a) Equal triangles on the same base, and on the same side of -it, are between the same parallels. — I. 39. (6) In what sense are these triangles equal ? (c) Prove that the straight line joining the...

...which must have preceded these convincing demonstrations. Let us suppose now that we begin with two equal triangles on the same base and on the same side of it, the larger pair of equal angles at the base being obtuse. Thus we have on the base AB the two triangles...

...equal in area (Euc. /. 38) 188 Ar.Sc. — Equal triangles on equal bases in the same straight line, and on the same side of it, are between the same parallels (Euc. I. 40) 188 Loci. L.6. — The locus of the vertex of a triangle of given area and standing upon...

...prove the converse of this proposition. 2. Equal triangles, on equal bases, in the same straight line, and on the same side of it, are between the same parallels. 3. If a straight line be divided into two equal and also into two unequal parts, the squares on the...

...part, BF is greater than BE ; therefore AC, CB are together greater than AD, DB. And, generally, of all triangles on the same base and on the same side of it which have equal vertical angles, the isosceles triangle is that which has the greatest perimeter,...

...Euclid's proof of Proposition VII of the First Book may be exhibited as a dilemma of this kind : ' If two triangles on the same base, and on the same side of it, have their conterminous sides equal, then two angles are both equal and unequal to each other ; but...

...point in the base is less than one of the equal sides of the triangle. Ex. 160. If ABC and ABD are two triangles on the same base and on the same side of it such that AC — BD and AD = BC, and if AC and BD intersect at 0, prove triangle AOB isosceles. Ex....