Pythagorean triples – Equal products of a leg and hypotenuse

The following integers

$a_1 \; = \; p^2 \; - \; q^2$
$b_1 \; = \; 2 \, p \, q$
$c_1 \; = \; p^2 \; + \; q^2$

$a_2 \; = \; r^2 \; - \; s^2$
$b_2 \; = \; 2 \, r \, s$
$c_2 \; = \; r^2 \; + \; s^2$

define two right triangles   $(a_1, \; b_1, \; c_1)$,    $(a_2, \; b_2, \; c_2)$.

Equal products of a leg and hypotenuse :

$(2 \, p \, q) \,(p^2 + q^2) \; = \; (2 \, r \, s) \,(r^2 + s^2)$

Here are some solutions:

(3504, 21172, 21460),    (7104, 7847, 10585)

$3504^2 + 21172^2 = 21460^2$ ….. $7104^2 + 7847^2 = 10585^2$

$3504 \times 21460 \; = \; 7104 \times 10585 \; = \; 75195840$

(278588, 14784, 278980),   (36977, 59136, 69745)

$278588^2 \; + \; 14784^2 \; = \; 278980^2$ ….. $36977^2 \; + \; 59136^2 \; = \; 69745^2$

$14784 \times 278980 \; = \; 59136 \times 69745 \; = \; 4124440320$

(5658912, 128466, 5660370),    (198512, 841266, 864370)

$5658912^2 + 128466^2 = 5660370^2$ ….. $198512^2 + 841266^2 = 864370^2$

$128466 \times 5660370 \; = \; 841266 \times 864370 \; = \; 727165092420$

(298734352, 1659264, 298738960),    (3530017, 22124544, 22404385)

$298734352^2 + 1659264^2 = 298738960^2$ ….. $3530017^2 + 22124544^2 = 22404385^2$

$1659264 \times 298738960 \; = \; 22124544 \times 22404385 \; = \; 495686801725440$

(6561804400, 12150750, 6561815650),    (119498176, 270018750, 295279426)

$6561804400^2 + 12150750^2 = 6561815650^2$ ….. $119498176^2 + 270018750^2 = 295279426^2$

$12150750 \times 6561815650 \; = \; 270018750 \times 295279426 \; = \; 79730981509237500$

(82533234132, 62053776, 82533257460),
(1400702617, 2057576256, 2489093785)

$82533234132^2 \; + \; 62053776^2 \; = \; 82533257460^2$
$1400702617^2 \; + \; 2057576256^2 \; = \; 2489093785^2$

$62053776 \times 82533257460 \; = \; 2057576256 \times 2489093785 \; = \; 5121500270973168960$

(705047366632, 246863274, 705047409850)
(10203455368, 11384759274, 15288009850)

$705047366632^2 \; + \; 246863274^2 \; = \; 705047409850$
$10203455368^2 \; + \; 11384759274^2 \; = \; 15288009850$

$246863274 \times 705047409850 \; = \; 11384759274 \times 15288009850$
$= \; 174050311920790848900$

(4532232204352, 817499136, 4532232278080)
(54860959873, 49941774336, 74188312705)

$4532232204352^2 \; + \; 817499136^2 \; = \; 4532232278080^2$
$54860959873^2 \; + \; 49941774336^2 \; = \; 74188312705^2$

$817499136 \times 4532232278080 \; = \; 49941774336 \times 74188312705$
$= \; 3705095971481711738880$

(23431108184712, 2352516534, 23431108302810)
(237358537192, 183666363894, 300120656410)

$23431108184712^2 \; + \; 2352516534^2 \; = \; 23431108302810^2$
$237358537192^2 \; + \; 183666363894^2 \; = \; 300120656410^2$

$2352516534 \times 23431108302810 \; = \; 183666363894 \times 300120656410$
$= \; 55122069692305203660540$

(28075678656192, 8566932018744, 29353637870280)
(394642839154432, 637209819576, 394643353589320)

$28075678656192^2 \; + \; 8566932018744^2 \; = \; 29353637870280^2$
$394642839154432^2 \; + \; 637209819576^2 \; = \; 394643353589320^2$

$8566932018744 \times 29353637870280 \; = \; 637209819576 \times 394643353589320$
$= \; 251470620137518169200528320$

(101969805870100, 6058806000, 101969806050100)
(870597970201, 588060600000, 1050597970201)

$101969805870100^2 \; + \; 6058806000^2 \; = \; 101969806050100^2$
$870597970201^2 \; + \; 588060600000^2 \; = \; 1050597970201^2$

$6058806000 \times 101969806050100 \; = \; 588060600000 \times 1050597970201$
$= \; 617815272715182180600000$