THEOREM 24. (Eucl. VI. 6.) Two triangles which have an equal angle formed by proportional sides, are similar to each other. Let A B C and D E F be two triangles having the angle B equal to the angle E, and let A B be to D E as B C is to E F. If A B be equal to D E, the side BC is equal to E F [Gen. Def. 28]: therefore the two triangles are equal [16], and consequently similar, to each other. If AB be not equal to DE, let AB be greater than DE [Gen. Def. 28]; then B C is greater than E F. Let the triangle DEF be applied to the triangle A B C, so that the vertex E coincide with the vertex B, and the side ED with the side BA; then, because the angle E is equal to the angle B[Hyp.], the side EF will coincide with the side BC. Let H and I be the points of B A and BC coinciding with the points D and F; then the side D F will B H A D E take a position HI. Because the lines HI and AC subtend proportionally the angle ABC [Hyp.], they are parallel to each other [II. 39]; hence the corresponding angles BHI and B A C are equal to each other [II. 20 i]; therefore the triangle ABC is similar to the triangle HBI [23]. But the triangle HBI is no other than the triangle D E F [Const.]; therefore the triangles A B C and D E F are similar to each other, W. W. T. B. D. Inversely If two triangles have two sides of the one proportional to two sides of the other, and if the third sides be not proportional to the first, the angles formed by the proportional sides are unequal. COROLLARY I. Two right-angled triangles having proportional perpendicular sides, are similar to each other. COROLLARY II. Two isosceles triangles which have an equal angle at the apex, are similar to each other. COROLLARY III. Two isosceles right-angled triangles are similar to each other. THEOREM 25. (Eucl. VI. 5.) Two triangles which have the three sides of the one proportional to the three sides of the other, are similar to each Let ABC and D E F be two triangles such that A B be to D E as BC is to E F and as A C is to D F. If the side A B be equal to the side DE, the side B C is equal to E F and AC to DF [Gen. Def. 28]; therefore the two triangles are equal [18], and consequently similar, to each other. If AB be not equal to DE, let AB be greater than DE; then BC is greater than E F [Gen. Def. 28]. Let BH be equal to ED; let BI be equal to EF, and let HI be joined then BA is to BH as BC is to BI, and as AC is to DF [Hyp.]. Because the lines HI and AC subtend proportionally the angle A B C, they are parallel to each other [II. 39]; therefore BA is to BH as AC is to HI [II. 40]: but BA is already to BH as AC is to DF [Hyp.]; then HI is equal to DF [xvii]; therefore the triangles HBI and D E F are equal [18], and consequently similar, to each other. Because the lines HI and AC are parallel to each other [Dem.], the corresponding angles BAC and BHI are equal to each other [II. 20 i]; therefore the triangles A B C and H BI are similar to each other [23]: but the triangle HBI is similar to the triangle DEF [Dem.]; therefore the triangles ABC and DEF are similar to each other, W.W.T. B. D. Inversely If two triangles have two sides of the one proportional to two sides of the other, and if the angles formed by these sides be not equal to each other, the third sides of the triangles are not proportional to the other sides. COROLLARY. Two isosceles triangles which have the base and one side of the one, proportional to the base and one side of the other, are similar to each other. Scholium. Equilateral triangles are similar to one another. THEOREM 26. (Eucl. VI. 7.) If two triangles have one angle of the one equal to one angle of the other, if the sides forming another angle be proportional, and if the third angle be acute in both triangles, the two triangles are similar to each other. Let A B C and D E F be two triangles having the angle B equal to the angle E; let AB be to D E as A C is to D F, and let the angles C and F be both acute. If the side A B be equal to the side DE, the side A C is equal to the side DF; therefore the two triangles are equal [20], and consequently similar, to each other. If AB be not equal to D E, let A B be greater than DE; let also BH be equal to D E and let HI be a parallel to AC: then HI is to AC as BH is to BA [II. 41]. Because D F is to A Cas D E is to A B [Hyp.], and D E is equal to BH [Const.], the line H I is to A C as DF is to A C [i]; therefore HI is equal to D F [xvi]. Because the corresponding angles BIH and BCA are equal to each other [II. 20 i] and the angle BCA is acute, the angle BIH is also acute [i]; there B I D other. (Eucl. VI. 7.) E fore the triangles HBI and DEF COROLLARY I. If two triangles have one angle of the one equal to one angle of the other, if the sides forming another angle be proportional, and if the third angle be obtuse in both triangles, the two triangles are similar to each COROLLARY II. Íf two triangles have one angle of the one equal to one angle of the other, and if the sides forming another angle be proportional, the third angle of each triangle is either equal or supplementary to the third angle of the other. THEOREM 27. If two right-angled triangles have the hypothenuse and one of the right sides of the one, respectively proportional to the hypothenuse and to one of the right sides of the other, the two triangles are similar to each other. Let A B C and DEF be two right-angled triangles, and let the hypothenuse BC be to the hypothenuse EF as the side B A is to the side E D. If B A be equal to ED, then BC is equal to EF; therefore the two triangles are equal [20], and consequently similar, to each other. B E If BA be not equal to ED, let B A be greater than ED; then BC will be greater than E F. Let BH be equal to ED, and let B I be equal to E F; let also HI be joined: then BA is to BH as BC is to BI[Hyp.]. Because the lines HI and AC subtend proportionally the angle ABC, they are parallel to each other [II. 38]; hence the corresponding angles B A C and B HI are equal to each other [II. 20 i]: therefore the triangle BHI is rightangled in H. Because the D right-angled triangles B A C and BHI have the angle B in common, they are similar to each other [23 ii]; because the right-angled triangles BHI and D E F have the hypothenuse BI and the side BH of the one respectively equal to the hypothenuse EF and the side ED of the other [Const.], they are equal [21], and consequently similar, to each other; therefore the triangles A B C and D E F are similar to each other, W. W. T. B. D. Scholium. Two isosceles right-angled triangles are similar to each other (24 iii). |