Como x, y e z são positivos, podemos dividir ambos os membros por ![[;sqrt{\frac{(x+y)(y+z)(z+x)}{x+y+z}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tQCC_DTUMinIXIrdeArXSXOKZPH6lFO4NELj82vONybOHY4Old2-9At1vG92wTu3SINFd0heYDwBToXoEDPZVjI0BmrQyP0jIwHfwPK78YxOumjIwvV6BH9zrXVJ_IYZPLlmK8WYl3UqhunVDSxNyUFpWaug=s0-d "sqrt{\frac{(x+y)(y+z)(z+x)}{x+y+z}}")
e obtermos:
e obtermos:![[;\sqrt{\frac{x(x+y+z)}{(x+y)(x+z)}}+\sqrt{\frac{y(x+y+z)}{(y+z)(y+x)}}+\sqrt{\frac{z(x+y+z)}{(z+x)(z+y)}}\geq2;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sRpuFiIAo0EqWFEB30yl9QlAqQnzeOFxtaUiOaWEJDMYymKzAkNTNszOpkw-5sI5vQFzWz7E2xLE1ivfAO4v6K2N5T_4WQose-ngc85Os5rMybHyYFyQ3AEeTN9XfSah8LgIwmNI0QU4GihpVjMj5TidOW84cATMVt0zOZLzqlHH_nj7ySCJlk-nHDiw7adwG12u6IpyUANaZ0-xKe21MxMw88_qMZrnRz8utiuOSGprKoUGb8XxGdy1s_ip0fT7DWH13t5jh4-WCdrVJn7qj4zrfNYX-7Zp0NDxMJFvXtixLMVkkP_nGTcaM8gvnqtP1-Qr5wXC2sxzyPMA=s0-d "\sqrt{\frac{x(x+y+z)}{(x+y)(x+z)}}+\sqrt{\frac{y(x+y+z)}{(y+z)(y+x)}}+\sqrt{\frac{z(x+y+z)}{(z+x)(z+y)}}\geq2")
(I)Mas se x, y e z são positivos, existe um triângulo com lados ![[;a=x+y;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_smV6yl8XFhto_y7Qllp5aslndsnbQKe-gPvVZH-nIQcDcQX83QKdpUH-x5wFAlUtXKjvbF3-2u2PW-T8am=s0-d "a=x+y")
, ![[;b=y+z;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u-VIntMCaa204B_3HD_ehrjQsntcCUEwDnJ30cLGIBMnB1UNuyjbs7SN03fVCt91ts6U4fJNKCkeSchEaG=s0-d "b=y+z")
e ![[;c=z+x;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vXOxumDuZAPa8FmmolvOs25K80nWDJ0OwO0Pbavan_y3l0PcyfkDSU7U3kZlpUO0QU88YOP7u0cGQALQf9=s0-d "c=z+x")
. Seja então ![[;s=x+y+z;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sH29Iiiw6NX1vCl3y6J9s26Y0MkQd-R5cnu4p3cqSADskEpbV-kdXw64ejRg6AgGhbHnpTNgVWjjE9WOm2NuA=s0-d "s=x+y+z")
, temos ![[;(x, y, z) = (s-b, s-c, s-a);]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_un938iQ9h57TlXSyho0Y0GyeTTxRGaTOnoQN4PUr1Ff-GMKcbbTGDayXHUHu30P6yr6w2OiFlSbp7IzIaxUqZDC4WdRYl57AbF0yiahEZdNI1fX70StnTkbi8roxpdLgIe8vgsCNBX=s0-d "(x, y, z) = (s-b, s-c, s-a)")
.
, e . Seja então , temos . Mas, do Teorema dos Cossenos:
![[;cos\gamma = \frac{a^{2}+b^{2}-c^{2}}{2ab};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_skdE5XzktT0ve8ra3Nj8DgWBlUL8MggLUVh7mBuQ24iEGUSMgQGvd1vaKZxMvGo-V-EUYLccWzPzJHY7e6yLMSMahFxeIatC1gf_zNAO6-6QBw5kwfskQjKoNUajGUnrPm9gf3--hHAHw_nbiJibMHXgAj-xtVOGDe4Zy9BW67nuknCg=s0-d "cos\gamma = \frac{a^{2}+b^{2}-c^{2}}{2ab}")
![[;=\frac{{(a+b)}^{2}-c^{2}}{2ab}-1;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sJq5QDuDaKnigMViZqb6BSwGWZ9xf8ur9x9sJHY49r0ZpB9fvcZHHRwkblytd_O6cF8zmqqm2cWNNhc1SSHb5uTaSCitnz0BKg2iNWdjEiBbkms71v_5ZLDVgJDw6-UItt-rtfmEps4pzb29O_-6Axw9KO0PWu=s0-d "=\frac{{(a+b)}^{2}-c^{2}}{2ab}-1")
![[;= \frac{(a+b+c)(a+b-c)}{2ab}-1;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sOrh4LqVjIeEuvmVDnIONdyNYa5E0CrNsWV1wbGYsnuQZAjLG7t5BOH97He9-O23TfVNR-NJtOO5UXSwIkrKlW75IyWjf6AgUuLLGWqb3Z3diyv-hmapN4cYdmu3Uy4HVteLNknBy65qfs=s0-d "= \frac{(a+b+c)(a+b-c)}{2ab}-1")
Como ![[;cos\gamma = 2{(\cos\frac{\gamma}{2})^{2}-1;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v9RC6XURCdflqrz2jA1ydt4XGlPXhWv-VyAol8d00N_2b1O9qQhAofrmI-tI9LP_77rWlWhcSh55Z8hTYeuqqTzORFg_3RaataM8AEBqCNtz46jwCAwlyxv0BF0Aha9Ceqf7uv783QTNyo0mV4xVifPr_EhgLZlZd6cDEajlT1HUiL=s0-d "cos\gamma = 2{(\cos\frac{\gamma}{2})^{2}-1")
:
:![[;\cos\frac{\gamma}{2} = \sqrt{\frac{s(s-c)}{ab}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_upOT8Mj5JG5IbhBYPAKRPCG8ptMFPVRcNcC3LDd-ZCoO7oJ8ijLdoE2ouX765Ce6J3pK7fNH8wgwxPvhz34LxDrRCN5GRvz1Xy9LLWrxWagpi19evivYZcyrbJ76ymr4ZCjV9_5uD0gMs9AliO9XoAdQ73DFEcbRkO3U5FNbSgWTe1ZBMpLSpabjFB0Jw=s0-d "\cos\frac{\gamma}{2} = \sqrt{\frac{s(s-c)}{ab}}")
![[;= \sqrt{\frac{y(x+y+z)}{(x+y)(y+z)}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_urrmqujxeM4jzeFSNpd04F_VLXfQQUbMlGgezzHo96TtoFkGxPuFi0BIhFv9jstvMuw6tIjB3XqZuLLzsk1ABPI52JK-S8R5vqHP8F9EyQY9OvBBdE5r63iQEwaPnGc_easlXfqWg1n1qh0qPCzY_j5u5WmgP8XN2Cqg=s0-d "= \sqrt{\frac{y(x+y+z)}{(x+y)(y+z)}}")
Analogamente, concluímos que:
![[;\cos(\alpha/2) = \sqrt{\frac{s(s-a)}{bc}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vji2pfoPgNd6ikjLAr49uG0Lw2gPMjX8Y57m8Gn9Xy6mXpsYtJMOARK_iMQMI8Dn3aqu2fHq_AbP_uf9l4joY6OYWnHX7pG5dOLIblvUNbQc_rHnEjcQyjiXAL5_evf_uA-pgqQAf41w4zewY_SIYf8bYSORLtolPPakstE-1jznM=s0-d "\cos(\alpha/2) = \sqrt{\frac{s(s-a)}{bc}}")
![[;=\sqrt{\frac{z(x+y+z)}{(y+z)(z+x)}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ug3cI2rnBWwXKdyFhO7V79G4vY5nobXRlPSTv9HFTWI-xMjvZ_11oIeJEERq-IGJDdJoaaiTZDVSlwzPnlxdhDEO_GF3aQHhmkguYtIJ2UlP0ZGK0ZAXHqGvEAmYXUdo3LV_tTPJnewL4ml5tWOeSk_5Zvo683=s0-d "=\sqrt{\frac{z(x+y+z)}{(y+z)(z+x)}}")
E:
![[;cos(\beta/2) = \sqrt{\frac{s(s-b)}{ac}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sFFZO1BwMMQMyugL2WnUXc959NZSHWLw60B3g1dHbOISx4wINiRdRbQszc2SKBRiPWrRr8RtOTt8rXwAoo6ejwge19U-yMd2s0hll7F6mlddbh6tWVMEzZkg12I4hyknjvwDKQXWwxDjvB-33CK9FnhngvphIDhrjZdFl0=s0-d "cos(\beta/2) = \sqrt{\frac{s(s-b)}{ac}}")
![[;=\sqrt{\frac{x(x+y+z)}{(x+y)(x+z)}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u4LJ2sO4wyF7-RRLvnjSeGxClRTVi-FABw948-PMrKN8LUJLB80xQE0wv5MnqIeBhby4gvnA04x9_B7wFOd-S9QPDKhxu6IzFMNr8ZYQqJ5k64hyd4fLwqP0Jppdd7BM52S-qAJA-LvE8KYnh1YEy9-z1J07hl=s0-d "=\sqrt{\frac{x(x+y+z)}{(x+y)(x+z)}}")
Substituindo em (I):
![[;\cos{\frac{\alpha}{2}} + \cos{\frac{\beta}{2}} + \cos{\frac{\gamma}{2}}=;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhSrySWa02LxCen9occyWwhEwrXpBk2JNqchnbJAJH4NwLoRwI4XbaqkRbSI6j7Jtr8xyC4KRPTofigULDFz1c6cWnaCDaF2zjfRayMVBSaIhIBEKsdSp_dmdWsMAlbPg1l4z4xETArP06WwtI3kE2ePrqNgkdOlqyJR6CA7VK6gUL1L-RIXyRyaNTyVbGHYuvKlJkq0K-Nkz3NayKw3I78W7HEzb4JRi_6C0MDmk6PD1XYegSF_MMeGuYUhXYTA=s0-d "\cos{\frac{\alpha}{2}} + \cos{\frac{\beta}{2}} + \cos{\frac{\gamma}{2}}=")
![[;\sqrt{\frac{x(x+y+z)}{(x+y)(x+z)}} + \sqrt{\frac{y(x+y+z)}{(y+x)(y+z)}} + \sqrt{\frac{z(x+y+z)}{(z+x)(z+y)}};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uKxW7XmrdlYTihN9FBoiF9RQf5Jn30HdcQQg9nKNrSKrr8K45m8f5QruvMS19rgW7dNEsfwLD_Mxt6dB04vwncefFV9xwDO1Jv9yCG2Vpct0sqb83D155-UhKSezdFAkQ6-Rs_UrATzJaVa79g3hRJlgbmfvphYp2o56pjh2gfJ7NLdYa3dttqQhclNMfQCSQfvDx5Zxo2DTi297YMjd2nNe3rEMtR6ERA-2u7nwjyElL2SEGMMQgg8GM6mmTwbctxEjyYKU1BCboNl-xtvgW9O9z6_4T3qa5frZHHWcUifvOAP9HwCvBCVtqPsyMO7povMrwJ2_fb3q7mL2oSjbke=s0-d "\sqrt{\frac{x(x+y+z)}{(x+y)(x+z)}} + \sqrt{\frac{y(x+y+z)}{(y+x)(y+z)}} + \sqrt{\frac{z(x+y+z)}{(z+x)(z+y)}}"){kind=small}
(*)Podemos ainda fazer as seguintes substituições:
Note que
(i.e. ângulos de um triângulo acutângulo) e:
(i.e. ângulos de um triângulo acutângulo) e:
Da desigualdade de Jordan (vide http://dadosdedeus.blogspot.com/2011/05/desigualdade-de-jordan.html), segue de imediato que:
[CQD](*) Existem várias outras formas de se provar essa desigualdade, como pela Desigualdade de Jensen, já que f(x) = cos(x) é uma função estritamente côncava no intervalo [0,π/2].
Notei que você utilizou em sua demonstração a desigualdade de jordan, já abordado em outra postagem, e isso mostra que a matemática não tem limites pois algo aparentemente complexo se torna simplificado apenas com o uso da ferramenta certa.
ResponderExcluirParabéns!
http://gigamatematica.blogspot.com
é verdade, Diego! Com a ferramenta certa podemos resolver uma grande quantidade de problemas de forma simples.
ResponderExcluirObrigado pelo comentário e volte sempre!