Let ABC, A'B'C' be equiangular triangles. Then shall AB: A'B' = BC: B'C' CA: C'A'. = Apply the triangle ABC to the triangle A'B'C', so that A may be on A', and AB on A'B', AC must then fall on A'C', since the angles A and A' are equal. The points B and C will fall somewhere on the lines. A'B', A'C'. If two triangles ABC, A'B'C', have the angle A equal to the angle A', and the sides about the equal angles proportional, so that AB : A'B' = AC: A'C', these triangles shall be equiangular. Apply the triangle ABC to the triangle A'B'C', so that the point A may be on A', and AB on A'B', AC must then fall on A'C' since the angles A and A' are equal. And since AB: A'B' AC: A'C', = BC is parallel to B'C'. [IV. 2. Therefore the remaining angles of the one triangle are equal to the remaining angles of the other each to each. [1. 8. THEOREM V. If in the triangles ABC, A'B'C', AB: A'B' BC: B'C' CA: C'A' = = On A'B' take A'D=AB, and through D draw DE parallel to B'C'. The triangle A'DE is equiangular to the triangle A'B'C'. Therefore A'D: A'B' DE: B'C'. = [I. 8. [IV. 3. But A'D being equal to AB, we have from hypothesis A'D: A'B' = BC: B'C'. Therefore the triangles A'DE and ABC are congruent and equiangular. But A'DE is equiangular to A'B'C' ; Relation of Areas of Figures. THEOREM VI. [I. 13. Triangles which have one angle of the one equal to one angle of the other, are to each other as the products of the sides containing the equal angle. Let the triangles ABC, A'BC' have equal angles at B. Then shall ABC: A'BC' = AB. BC: A'B.BC'. Place the triangles so that the sides containing the equal angle in the one may be upon those containing the equal angle in the other; and let fall the perpendiculars AD, A'D'. triangle ABC: triangle A'BC' = AD. BC: A'D' . BC'. But Therefore AD: A'D' AB: A'B. = [IV. 2. triangle ABC: triangle A'BC' = AB. BC: A'B.BC'. COR. 1. Parallelograms which have one angle of the one equal to one angle of the other are to one another as the products of the sides containing the equal angle. then COR. 2. If the triangle ABC the triangle A'BC', A = B C B A' AB. BC= A'B. BC', and therefore AB: A'B=BC': BC. i.e. the triangles ABC, A'BC' have their sides about the equal angles reciprocally proportional. COR. 3. If AB: A'BBC': BC, AB.BC- A'B. BC', and therefore the triangle ABC= triangle A'BC'. Or if ABC, A'BC' have their sides about their equal angles reciprocally proportional, these triangles are equal. DEF. Similar figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals. AADD It has been shewn that equiangular triangles have their sides about their equal angles proportionals; triangles therefore which are equiangular are necessarily similar, but this is not the case with other figures. A square and an oblong are equiangular but not similar, and parallelograms equiangular but not similar are of common occurrence. |