If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon will be equiangular.
如果一个等边五边形中有三个角相等（无论是否按顺序），则该五边形是等角的。
等边五边形 ABCDE，顶点 A 在最上方，B、E 为上肩，C、D 为下底。证明中连接对角线 AC、BE 与 DF——其中 F 是 AC 与 BE 的内交点；当连接 BD 时，BD、BE 构造出等量角的对比三角形。
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For in the equilateral pentagon ABCDE let, first, three angles taken in order, those at A, B, C, be equal to one another; I say that the pentagon ABCDE is equiangular. For let AC, BE, FD be joined. Now, since the two sides CB, BA are equal to the two sides BA, AE respectively, and the angle CBA is equal to the angle BAE, therefore the base AC is equal to the base BE, the triangle ABC is equal to the triangle ABE, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend, [I. 4] that is, the angle BCA to the angle BEA, and the angle ABE to the angle CAB; hence the side AF is also equal to the side BF. [I. 6] But the whole AC was also proved equal to the whole BE; therefore the remainder FC is also equal to the remainder FE.
首先考虑三个角按顺序相等的情况。连接AC、BE、FD。由边角边定理，三角形ABC与ABE全等，得AC=BE，且角BCA=角BEA，角ABE=角CAB。
But CD is also equal to DE. Therefore the two sides FC, CD are equal to the two sides FE, ED; and the base FD is common to them; therefore the angle FCD is equal to the angle FED. [I. 8] But the angle BCA was also proved equal to the angle AEB; therefore the whole angle BCD is also equal to the whole angle AED. But, by hypothesis, the angle BCD is equal to the angles at A, B; therefore the angle AED is also equal to the angles at A, B.
由等角对等边，AF=BF；又AC=BE，故FC=FE。结合CD=DE，由边边边定理得三角形FCD与FED全等，故角FCD=角FED。
Similarly we can prove that the angle CDE is also equal to the angles at A, B, C; therefore the pentagon ABCDE is equiangular. Next, let the given equal angles not be angles taken in order, but let the angles at the points A, C, D be equal; I say that in this case too the pentagon ABCDE is equiangular. For let BD be joined. Then, since the two sides BA, AE are equal to the two sides BC, CD, and they contain equal angles, therefore the base BE is equal to the base BD, the triangle ABE is equal to the triangle BCD, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend; [I. 4] therefore the angle AEB is equal to the angle CDB.
由角BCA=角AEB，得角BCD=角AED。由假设角BCD等于角A、B，故角AED也等于角A、B。同理可证角CDE等于角A、B、C，因此五边形等角。
But the angle BED is also equal to the angle BDE, since the side BE is also equal to the side BD. [I. 5] Therefore the whole angle AED is equal to the whole angle CDE. But the angle CDE is, by hypothesis, equal to the angles at A, C; therefore the angle AED is also equal to the angles at A, C. For the same reason the angle ABC is also equal to the angles at A, C, D.
其次考虑三个角不按顺序相等的情况，设角A、C、D相等。连接BD。由边角边定理，三角形ABE与BCD全等，得BE=BD，角AEB=角CDB。由等边对等角，角BED=角BDE，故角AED=角CDE。由假设角CDE等于角A、C，得角AED等于角A、C。同理角ABC等于角A、C、D，故五边形等角。