In Fig. 1, three semicircles are made outward with three sides of RT △ ABC as diameters, and their areas are expressed by S1, S2 and S3 respectively. It is not difficult to prove that S1 = S2 + S3. (1) (1) As shown in Figure 2, take the three sides of RT △ ABC as the sides and make three squares outward, and their areas are represented by S1, S2 and S3 respectively. Write down their relationship. (it is not necessary to prove) (2) As shown in Figure 3, take the three sides of RT △ ABC as the edges and make regular triangles outward, and their areas are represented by S1, S2 and S3 respectively. Determine their relationship and prove it; (3) If three general triangles are made outward with three sides of RT △ ABC, and their areas are expressed by S1, S2 and S3 respectively, what conditions should be satisfied for the triangle to have the same relationship with (2) between S1, S2 and S3?

In Fig. 1, three semicircles are made outward with three sides of RT △ ABC as diameters, and their areas are expressed by S1, S2 and S3 respectively. It is not difficult to prove that S1 = S2 + S3. (1) (1) As shown in Figure 2, take the three sides of RT △ ABC as the sides and make three squares outward, and their areas are represented by S1, S2 and S3 respectively. Write down their relationship. (it is not necessary to prove) (2) As shown in Figure 3, take the three sides of RT △ ABC as the edges and make regular triangles outward, and their areas are represented by S1, S2 and S3 respectively. Determine their relationship and prove it; (3) If three general triangles are made outward with three sides of RT △ ABC, and their areas are expressed by S1, S2 and S3 respectively, what conditions should be satisfied for the triangle to have the same relationship with (2) between S1, S2 and S3?

(2) the area formula of equilateral triangle: S = √ 3 / 4 * a ^ 2 (where a is the side length of equilateral triangle).. S1 = √ 3 / 4C ^ 2, S2 = √ 3 / 4A ^ 2, S3 = √ 3 / 4B ^ 2, ∵ {C = 90 °, a ^ 2 + B ^ 2 = C ^ 2, S2 + S3 = √ 3 / 4 (a ^ 2 + B ^ 2) = √ 3 / 4C ^ 2