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When you enlarge the triangle by scale factor a and b, the lengths become ab, a^2 and ac for the first triangle and ab, b^2 and bc for the second triangle. The angles stay the same as it has been increased by a scale factor. When the 2 are joined together, the new larger triangle gets the angle x from the second triangle and the angle y from the first triangle. This means the last angle must be 90 degrees therefore it is similar to the original triangle. The triangle has sides ac, bc and a^2 + b^2. We also know that it has height ab if a^2+b^2 is the base as the 2 triangles were joined together at the line with length ab. Because of this there are 2 ways to find the area using the formula area = base*height. ac*bc and ab(a^2+b^2). ac*bc works as they meet at a 90 degree so bc is perpendicular to ac meaning bc is the height for ac as a base. This means abc^2=(a^3)b+a(b^3). Then if we divide both sides by ab we get a^2+b^2=c^2.