Hence it will be observed that the equality of the angles involves the similarity of the triangles; and the additional equality of a pair of corresponding sides involves the identity of the triangles. THEOREM 7. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, then will the triangles be similar. Let the triangles ABC, DEF have the angles at B and E equal, and let BA : BC :: ED: EF, then will the triangles be similar. Conceive the angle E placed on the equal angle B, then D and F will fall as at D' and F' on the sides BA, BC, BA: BC ED : EF, and since therefore BA: BD :: BC: BF", and therefore D'F' is parallel to AC, TH. 4. COR. 3. is, D and F, are Hence the trian It will be observed that this theorem is a generalization of Book 1. Theorem 17. If two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another, the triangles will be equal in all respects. THEOREM 8. If the sides about each of the angles of two triangles are proportionals, the triangles will be similar. Let ABC, DEF be two triangles which have their sides. Conceive a triangle equiangular to ABC applied to EF, on the opposite side of the base EF, so that the angles FEG, EFG are equal to B and C respectively. Then the triangle GEF is equiangular to ABC, and therefore similar to it, and the triangle DEF is therefore equiangular to GEF, and therefore also to ABC. Therefore the triangle DEF is similar to the triangle ABC. This theorem is a generalization of Book 1. Theorem 18. If the three sides of one triangle are respectively equal to the three sides of another, these triangles will be equal in all respects. THEOREM 9. If two triangles have the sides about an angle of the one triangle proportional to the sides about an angle of the other, and have also the angle opposite that which is not the less of the two sides of the one equal to the corresponding angle of the other, these triangles will be similar. Let ABC, DEF be the two triangles, in which BA: AC :: ED : DF, A A and let AC be not less than AB, and DF therefore not less than DE, and also let the angle B = the angle E. Then shall the triangles be similar. Cut off BD – ED, and draw to BC an oblique D'F' parallel to AC. Then by similar triangles BD'F' and BAC, BD : D'F' :: BA : AC, and since DF is not less than DE, the two triangles BD'F', EDF are equal in all respects by Book 1. Theorem 19, that is, the triangle EDF is equiangular to the triangle BD'F', and therefore also the triangle EDF is equiangular to ABC, and therefore similar to ABC. This theorem is a generalization of Book 1. Theorem 19. THEOREM IO. Similar polygons can be divided into the same number of similar triangles. Let ABCDE, PQRST be similar polygons, that is, let the angles A, B, C... of the one equal the angles P, Q, R... of the other respectively, and let the sides which contain the equal angles be proportional. That is, let EA: AB :: TP : PQ, and similarly for the sides about the other angles. Take any point O within the polygon ABCDE. Join OA, OE. Make the angles TPO, PTƠ equal to EAO, AEO respectively. Then if OB, OC... O'Q, O'R... be drawn, the polygons will be divided into the same number of similar triangles. For join OB, O'Q. but Then since the triangles OAE, O’PT are equiangular, .. and since the angle BAE=the angle QPT, and OAE=O'PT, ... the angle OAB = the angle OPQ; .'. the triangles OAB, O'PQ are similar; = and therefore OBA O'QP, and OB : BA :: O'Q : PQ; in the same way it may be shewn that if OC, O'R are joined, the triangle OBC is similar to the triangle O'QR. Hence the polygons can be divided into the same number of similar triangles. The points O, O' are then called homologous points, and the lines OB, O'Q; OA, OP, &c., homologous lines. COR. I. The perimeters of similar polygons have to one another the ratio of the homologous sides of the polygon. |