Mathematical_Recreations-Kraitchik-2e

106 Mathematical Recreationa
OA = OA'= xu = 4abcd,
DB = DB' = yv = (a 2 - b2 )(C 2 - d2),
DC = ~C' = yu = 2cd(a2 - b 2 ),
OD = OD' = xv = 2ab(c 2 - d 2 ).
Then AB = A'B', AB' = A'B, CD = C'D' and CD' = C'D
are formed from these by taking sums and differences. (In
a figure based on different right triangles the relative positions
of A and B may be interchanged, as may those of C and
D.) Also,
AC = AC' = A'C = A'C' = zu = 2cd(a + 2 b 2 ),
AD = AD'= A'D = A'D' = xw = 2ab(c + 2 d 2 ),
BC = BC' = B'C = B'C' = yw = (a 2 - b 2 )(C + 2 d2),
BD = BD' = B'D = B'D' = zv = (a + 2 b 2 )(C 2 - d 2 ).
The scalene triangles are:
ABC = ABC' = A'B'C = A'B'C',
AB'C = AB'C' = A'BC = A'BC',
ABD = ABD' = A'B'D = A'B'D',
AB'D = AB'D' = A'BD = A'BD',
CDA = CDA' = C'D'A = C'D'A',
CD'A = CD'A' = C'DA = C'DA',
CDB = CDB' = C'D'B = C'D'B',
CD'B = CD'B' = C'DB = C'DB'.
The isosceles triangles are:
AA'C = AA'C',
AA'D = AA'D',
BB'C = BB'C',
BB'D = BB'D',
to which the following are respectively similar:
DD'B = DD'B',
CC'B = CC'B',
DD'A = DD'A',
CC'A = CC'A'.