三角形内点幂和极值问题

数学星空

同理：对于\(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\)的最小值？

我们可以得到：

\(a^2x^6y^2z^2-x^6y^6-x^6y^4z^2-x^6y^2z^4-x^6z^6+y^6z^6=0\)

\(b^2x^2y^6z^2-x^6y^6+x^6z^6-x^4y^6z^2-x^2y^6z^4-y^6z^6=0\)

\(c^2x^2y^2z^6+x^6y^6-x^6z^6-x^4y^2z^6-x^2y^4z^6-y^6z^6=0\)

数学星空

看来，边长平方后的结果要简洁很多，11#的消元结果为：

1. a^4b^{14}c^{14}+a^4b^{12}c^{12}(4a^2-7b^2-7c^2)x^2+a^2b^{10}c^{10}(6a^6-22a^4b^2-22a^4c^2+11a^2b^4+45a^2b^2c^2+11a^2c^4+2b^6-2b^4c^2-2b^2c^4+2c^6)x^4+a^2b^8c^8(4a^8-26a^6b^2-26a^6c^2+18a^4b^4+117a^4b^2c^2+18a^4c^4+10a^2b^6-66a^2b^4c^2-66a^2b^2c^4+10a^2c^6-10b^8-4b^6c^2+28b^4c^4-4b^2c^6-10c^8)x^6+b^6c^6(a^{12}-14a^{10}b^2-14a^{10}c^2+5a^8b^4+114a^8b^2c^2+5a^8c^4+18a^6b^6-90a^6b^4c^2-90a^6b^2c^4+18a^6c^6-15a^4b^8-40a^4b^6c^2+113a^4b^4c^4-40a^4b^2c^6-15a^4c^8+4a^2b^{10}+59a^2b^8c^2-67a^2b^6c^4-67a^2b^4c^6+59a^2b^2c^8+4a^2c^{10}+b^{12}-b^{10}c^2-b^8c^4+2b^6c^6-b^4c^8-b^2c^{10}+c^{12})x^8-b^4c^4(3a^{12}b^2+3a^{12}c^2+6a^{10}b^4-53a^{10}b^2c^2+6a^{10}c^4-3a^8b^6+32a^8b^4c^2+32a^8b^2c^4-3a^8c^6-28a^6b^8+55a^6b^6c^2-83a^6b^4c^4+55a^6b^2c^6-28a^6c^8+29a^4b^{10}-60a^4b^8c^2+76a^4b^6c^4+76a^4b^4c^6-60a^4b^2c^8+29a^4c^{10}-10a^2b^{12}+18a^2b^{10}c^2+62a^2b^8c^4-220a^2b^6c^6+62a^2b^4c^8+18a^2b^2c^{10}-10a^2c^{12}+3b^{14}+5b^{12}c^2-13b^{10}c^4+5b^8c^6+5b^6c^8-13b^4c^{10}+5b^2c^{12}+3c^{14})x^{10}-b^2c^2(3a^{12}b^4-9a^{12}b^2c^2+3a^{12}c^4+10a^{10}b^6-10a^{10}b^4c^2-10a^{10}b^2c^4+10a^{10}c^6-35a^8b^8-19a^8b^6c^2-68a^8b^4c^4-19a^8b^2c^6-35a^8c^8+12a^6b^{10}+168a^6b^8c^2+154a^6b^6c^4+154a^6b^4c^6+168a^6b^2c^8+12a^6c^{10}+29a^4b^{12}-187a^4b^{10}c^2-39a^4b^8c^4-327a^4b^6c^6-39a^4b^4c^8-187a^4b^2c^{10}+29a^4c^{12}-22a^2b^{14}+82a^2b^{12}c^2+12a^2b^{10}c^4+36a^2b^8c^6+36a^2b^6c^8+12a^2b^4c^{10}+82a^2b^2c^{12}-22a^2c^{14}+3b^{16}-25b^{14}c^2+4b^{12}c^4+37b^{10}c^6-38b^8c^8+37b^6c^{10}+4b^4c^{12}-25b^2c^{14}+3c^{16})x^{12}+(-4a^{12}b^6+9a^{12}b^4c^2+9a^{12}b^2c^4-4a^{12}c^6+8a^{10}b^8+26a^{10}b^6c^2+39a^{10}b^4c^4+26a^{10}b^2c^6+8a^{10}c^8+4a^8b^{10}-133a^8b^8c^2-121a^8b^6c^4-121a^8b^4c^6-133a^8b^2c^8+4a^8c^{10}-16a^6b^{12}+60a^6b^{10}c^2+206a^6b^8c^4+340a^6b^6c^6+206a^6b^4c^8+60a^6b^2c^{10}-16a^6c^{12}+4a^4b^{14}+159a^4b^{12}c^2-361a^4b^{10}c^4-66a^4b^8c^6-66a^4b^6c^8-361a^4b^4c^{10}+159a^4b^2c^{12}+4a^4c^{14}+8a^2b^{16}-150a^2b^{14}c^2+275a^2b^{12}c^4+18a^2b^{10}c^6-270a^2b^8c^8+18a^2b^6c^{10}+275a^2b^4c^{12}-150a^2b^2c^{14}+8a^2c^{16}-4b^{18}+29b^{16}c^2-79b^{14}c^4+39b^{12}c^6+15b^{10}c^8+15b^8c^{10}+39b^6c^{12}-79b^4c^{14}+29b^2c^{16}-4c^{18})x^{14}+(12a^{12}b^4+9a^{12}b^2c^2+12a^{12}c^4-44a^{10}b^6+39a^{10}b^4c^2+39a^{10}b^2c^4-44a^{10}c^6+24a^8b^8-34a^8b^6c^2-47a^8b^4c^4-34a^8b^2c^6+24a^8c^8+40a^6b^{10}-74a^6b^8c^2+105a^6b^6c^4+105a^6b^4c^6-74a^6b^2c^8+40a^6c^{10}-4a^4b^{12}-131a^4b^{10}c^2+245a^4b^8c^4-340a^4b^6c^6+245a^4b^4c^8-131a^4b^2c^{10}-4a^4c^{12}-60a^2b^{14}+307a^2b^{12}c^2-476a^2b^{10}c^4+313a^2b^8c^6+313a^2b^6c^8-476a^2b^4c^{10}+307a^2b^2c^{12}-60a^2c^{14}+32b^{16}-116b^{14}c^2+154b^{12}c^4-32b^{10}c^6-60b^8c^8-32b^6c^{10}+154b^4c^{12}-116b^2c^{14}+32c^{16})x^{16}+(12a^{12}b^2+12a^{12}c^2+32a^{10}b^4+43a^{10}b^2c^2+32a^{10}c^4-180a^8b^6-67a^8b^4c^2-67a^8b^2c^4-180a^8c^6+192a^6b^8+298a^6b^6c^2+228a^6b^4c^4+298a^6b^2c^6+192a^6c^8-124a^4b^{10}-286a^4b^8c^2-290a^4b^6c^4-290a^4b^4c^6-286a^4b^2c^8-124a^4c^{10}+160a^2b^{12}-245a^2b^{10}c^2+420a^2b^8c^4-46a^2b^6c^6+420a^2b^4c^8-245a^2b^2c^{10}+160a^2c^{12}-92b^{14}+245b^{12}c^2-275b^{10}c^4+74b^8c^6+74b^6c^8-275b^4c^{10}+245b^2c^{12}-92c^{14})x^{18}+(16a^{12}+16a^{10}b^2+16a^{10}c^2+76a^8b^4+113a^8b^2c^2+76a^8c^4-304a^6b^6-212a^6b^4c^2-212a^6b^2c^4-304a^6c^6+248a^4b^8+118a^4b^6c^2+764a^4b^4c^4+118a^4b^2c^6+248a^4c^8-192a^2b^{10}+364a^2b^8c^2-544a^2b^6c^4-544a^2b^4c^6+364a^2b^2c^8-192a^2c^{10}+140b^{12}-319b^{10}c^2+260b^8c^4-114b^6c^6+260b^4c^8-319b^2c^{10}+140c^{12})x^{20}+(80a^{10}-48a^6b^4+32a^6b^2c^2-48a^6c^4-48a^4b^6+16a^4b^4c^2+16a^4b^2c^4-48a^4c^6+160a^2b^8-560a^2b^6c^2+720a^2b^4c^4-560a^2b^2c^6+160a^2c^8-208b^{10}+400b^8c^2-128b^6c^4-128b^4c^6+400b^2c^8-208c^{10})x^{22}+(256a^8-340a^6b^2-340a^6c^2+252a^4b^4+352a^4b^2c^2+252a^4c^4-268a^2b^6+28a^2b^4c^2+28a^2b^2c^4-268a^2c^6+292b^8-424b^6c^2+312b^4c^4-424b^2c^6+292c^8)x^{24}+(336a^6-480a^4b^2-480a^4c^2+384a^2b^4+144a^2b^2c^2+384a^2c^4-240b^6+96b^4c^2+96b^2c^4-240c^6)x^{26}+(144a^4-144a^2b^2-144a^2c^2+144b^4-144b^2c^2+144c^4)x^{28}=0

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数学星空

对于\({x^n}+{y^n}+{z^n}\)的最小值：

当\(n=3\)时

\(a^2yz+x^4-y^4-y^3z-yz^3-z^4=0\)

\(b^2xz-x^4-x^3z-xz^3+y^4-z^4=0\)

\(c^2xy-x^4-x^3y-xy^3-y^4+z^4=0\)

消元结果为：

1. a^8b^2c^2(a^2-2b^2-2c^2)^2(a^4-2a^2b^2-2a^2c^2+2b^4+2c^4)^2+a^2(a^{20}-9a^{18}b^2-9a^{18}c^2+36a^{16}b^4+80a^{16}b^2c^2+36a^{16}c^4-85a^{14}b^6-300a^{14}b^4c^2-300a^{14}b^2c^4-85a^{14}c^6+133a^{12}b^8+630a^{12}b^6c^2+920a^{12}b^4c^4+630a^{12}b^2c^6+133a^{12}c^8-152a^{10}b^{10}-811a^{10}b^8c^2-1413a^{10}b^6c^4-1413a^{10}b^4c^6-811a^{10}b^2c^8-152a^{10}c^{10}+148a^8b^{12}+649a^8b^{10}c^2+1146a^8b^8c^4+1354a^8b^6c^6+1146a^8b^4c^8+649a^8b^2c^{10}+148a^8c^{12}-140a^6b^{14}-324a^6b^{12}c^2-432a^6b^{10}c^4-480a^6b^8c^6-480a^6b^6c^8-432a^6b^4c^{10}-324a^6b^2c^{12}-140a^6c^{14}+116a^4b^{16}+103a^4b^{14}c^2+74a^4b^{12}c^4-63a^4b^{10}c^6-76a^4b^8c^8-63a^4b^6c^{10}+74a^4b^4c^{12}+103a^4b^2c^{14}+116a^4c^{16}-64a^2b^{18}-16a^2b^{16}c^2-16a^2b^{14}c^4+64a^2b^{12}c^6+32a^2b^{10}c^8+32a^2b^8c^{10}+64a^2b^6c^{12}-16a^2b^4c^{14}-16a^2b^2c^{16}-64a^2c^{18}+16b^{20}-16b^{18}c^2-48b^{14}c^6+48b^{12}c^8+48b^8c^{12}-48b^6c^{14}-16b^2c^{18}+16c^{20})x^2+(17a^{20}-128a^{18}b^2-128a^{18}c^2+420a^{16}b^4+840a^{16}b^2c^2+420a^{16}c^4-793a^{14}b^6-2279a^{14}b^4c^2-2279a^{14}b^2c^4-793a^{14}c^6+980a^{12}b^8+3336a^{12}b^6c^2+4433a^{12}b^4c^4+3336a^{12}b^2c^6+980a^{12}c^8-949a^{10}b^{10}-2857a^{10}b^8c^2-3793a^{10}b^6c^4-3793a^{10}b^4c^6-2857a^{10}b^2c^8-949a^{10}c^{10}+923a^8b^{12}+1568a^8b^{10}c^2+970a^8b^8c^4+895a^8b^6c^6+970a^8b^4c^8+1568a^8b^2c^{10}+923a^8c^{12}-855a^6b^{14}-651a^6b^{12}c^2+393a^6b^{10}c^4+773a^6b^8c^6+773a^6b^6c^8+393a^6b^4c^{10}-651a^6b^2c^{12}-855a^6c^{14}+529a^4b^{16}+92a^4b^{14}c^2-383a^4b^{12}c^4-98a^4b^{10}c^6-136a^4b^8c^8-98a^4b^6c^{10}-383a^4b^4c^{12}+92a^4b^2c^{14}+529a^4c^{16}-147a^2b^{18}+159a^2b^{16}c^2+159a^2b^{14}c^4+147a^2b^{12}c^6-318a^2b^{10}c^8-318a^2b^8c^{10}+147a^2b^6c^{12}+159a^2b^4c^{14}+159a^2b^2c^{16}-147a^2c^{18}+3b^{20}-3b^{18}c^2-9b^{14}c^6+9b^{12}c^8+9b^8c^{12}-9b^6c^{14}-3b^2c^{18}+3c^{20})x^4+(120a^{18}-721a^{16}b^2-721a^{16}c^2+1830a^{14}b^4+3343a^{14}b^2c^2+1830a^{14}c^4-2583a^{12}b^6-5877a^{12}b^4c^2-5877a^{12}b^2c^4-2583a^{12}c^6+2488a^{10}b^8+4544a^{10}b^6c^2+4923a^{10}b^4c^4+4544a^{10}b^2c^6+2488a^{10}c^8-2390a^8b^{10}-1155a^8b^8c^2+1330a^8b^6c^4+1330a^8b^4c^6-1155a^8b^2c^8-2390a^8c^{10}+2490a^6b^{12}-308a^6b^{10}c^2-3328a^6b^8c^4-3172a^6b^6c^6-3328a^6b^4c^8-308a^6b^2c^{10}+2490a^6c^{12}-1767a^4b^{14}+714a^4b^{12}c^2+1886a^4b^{10}c^4-641a^4b^8c^6-641a^4b^6c^8+1886a^4b^4c^{10}+714a^4b^2c^{12}-1767a^4c^{14}+560a^2b^{16}-831a^2b^{14}c^2-907a^2b^{12}c^4+347a^2b^{10}c^6+1662a^2b^8c^8+347a^2b^6c^{10}-907a^2b^4c^{12}-831a^2b^2c^{14}+560a^2c^{16}-27b^{18}+27b^{16}c^2+27b^{14}c^4+27b^{12}c^6-54b^{10}c^8-54b^8c^{10}+27b^6c^{12}+27b^4c^{14}+27b^2c^{16}-27c^{18})x^6+(399a^{16}-1772a^{14}b^2-1772a^{14}c^2+3018a^{12}b^4+4768a^{12}b^2c^2+3018a^{12}c^4-2700a^{10}b^6-2248a^{10}b^4c^2-2248a^{10}b^2c^4-2700a^{10}c^6+2695a^8b^8-4425a^8b^6c^2-6413a^8b^4c^4-4425a^8b^2c^6+2695a^8c^8-4176a^6b^{10}+6088a^6b^8c^2+6316a^6b^6c^4+6316a^6b^4c^6+6088a^6b^2c^8-4176a^6c^{10}+3912a^4b^{12}-4570a^4b^{10}c^2-1554a^4b^8c^4+5520a^4b^6c^6-1554a^4b^4c^8-4570a^4b^2c^{10}+3912a^4c^{12}-1472a^2b^{14}+3128a^2b^{12}c^2+1572a^2b^{10}c^4-3372a^2b^8c^6-3372a^2b^6c^8+1572a^2b^4c^{10}+3128a^2b^2c^{12}-1472a^2c^{14}+96b^{16}-135b^{14}c^2-147b^{12}c^4+51b^{10}c^6+270b^8c^8+51b^6c^{10}-147b^4c^{12}-135b^2c^{14}+96c^{16})x^8+(477a^{14}-878a^{12}b^2-878a^{12}c^2-1050a^{10}b^4-2084a^{10}b^2c^2-1050a^{10}c^4+2360a^8b^6+8750a^8b^4c^2+8750a^8b^2c^4+2360a^8c^6+1733a^6b^8-9335a^6b^6c^2-1226a^6b^4c^4-9335a^6b^2c^6+1733a^6c^8-5040a^4b^{10}+8485a^4b^8c^2-9285a^4b^6c^4-9285a^4b^4c^6+8485a^4b^2c^8-5040a^4c^{10}+2620a^2b^{12}-7781a^2b^{10}c^2+2426a^2b^8c^4+5278a^2b^6c^6+2426a^2b^4c^8-7781a^2b^2c^{10}+2620a^2c^{12}-222b^{14}+519b^{12}c^2+243b^{10}c^4-540b^8c^6-540b^6c^8+243b^4c^{10}+519b^2c^{12}-222c^{14})x^{10}+(-137a^{12}+2136a^{10}b^2+2136a^{10}c^2-5998a^8b^4-4981a^8b^2c^2-5998a^8c^4+5180a^6b^6+626a^6b^4c^2+626a^6b^2c^4+5180a^6c^6+111a^4b^8+10a^4b^6c^2+12356a^4b^4c^4+10a^4b^2c^6+111a^4c^8-1628a^2b^{10}+7770a^2b^8c^2-4282a^2b^6c^4-4282a^2b^4c^6+7770a^2b^2c^8-1628a^2c^{10}+336b^{12}-1239b^{10}c^2+546b^8c^4+858b^6c^6+546b^4c^8-1239b^2c^{10}+336c^{12})x^{12}+(-38a^{10}-355a^8b^2-355a^8c^2+2088a^6b^4+6971a^6b^2c^2+2088a^6c^4-2974a^4b^6-8575a^4b^4c^2-8575a^4b^2c^4-2974a^4c^6+1294a^2b^8-3703a^2b^6c^2-198a^2b^4c^4-3703a^2b^2c^6+1294a^2c^8-15b^{10}+918b^8c^2-1191b^6c^4-1191b^4c^6+918b^2c^8-15c^{10})x^{14}+(9a^8-132a^6b^2-132a^6c^2+78a^4b^4-1353a^4b^2c^2+78a^4c^4+204a^2b^6+4176a^2b^4c^2+4176a^2b^2c^4+204a^2c^6-159b^8+186b^6c^2+738b^4c^4+186b^2c^6-159c^8)x^{16}+(27a^4b^2+27a^4c^2-54a^2b^4-342a^2b^2c^2-54a^2c^4+27b^6-531b^4c^2-531b^2c^4+27c^6)x^{18}+81b^2c^2x^{20}=0

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当\(n=4\)时

\(a^2y^2z^2+x^6-y^6-y^4z^2-y^2z^4-z^6=0\)

\(b^2x^2z^2-x^6-x^4z^2-x^2z^4+y^6-z^6=0\)

\(c^2x^2y^2-x^6-x^4y^2-x^2y^4-y^6+z^6=0\)

消元结果为：

\(a^4b^2c^2(a^2-2b^2-2c^2)(a^4-a^2b^2-a^2c^2+b^4-b^2c^2+c^4)+(a^{12}-4a^{10}b^2-4a^{10}c^2+7a^8b^4+15a^8b^2c^2+7a^8c^4-9a^6b^6-18a^6b^4c^2-18a^6b^2c^4-9a^6c^6+8a^4b^8+11a^4b^6c^2+10a^4b^4c^4+11a^4b^2c^6+8a^4c^8-5a^2b^{10}+a^2b^8c^2-4a^2b^6c^4-4a^2b^4c^6+a^2b^2c^8-5a^2c^{10}+2b^{12}-4b^{10}c^2+6b^8c^4-8b^6c^6+6b^4c^8-4b^2c^{10}+2c^{12})x^2+(7a^{10}-19a^8b^2-19a^8c^2+27a^6b^4+53a^6b^2c^2+27a^6c^4-29a^4b^6-43a^4b^4c^2-43a^4b^2c^4-29a^4c^6+20a^2b^8+26a^2b^6c^2-8a^2b^4c^4+26a^2b^2c^6+20a^2c^8-10b^{10}+6b^8c^2-4b^6c^4-4b^4c^6+6b^2c^8-10c^{10})x^4+(25a^8-37a^6b^2-37a^6c^2+39a^4b^4+43a^4b^2c^2+39a^4c^4-25a^2b^6-35a^2b^4c^2-35a^2b^2c^4-25a^2c^6+10b^8+20b^6c^2-12b^4c^4+20b^2c^6+10c^8)x^6+(48a^6-48a^4b^2-48a^4c^2+42a^2b^4-36a^2b^2c^2+42a^2c^4-6b^6+6b^4c^2+6b^2c^4-6c^6)x^8+(36a^4-36a^2b^2-36a^2c^2+36b^4-36b^2c^2+36c^4)x^{10}=0\)

当\(n=5\)时

\(a^2y^3z^3+x^8-y^8-y^5z^3-y^3z^5-z^8=0\)

\(b^2x^3z^3-x^8-x^5z^3-x^3z^5+y^8-z^8=0\)

\(c^2x^3y^3-x^8-x^5y^3-x^3y^5-y^8+z^8=0\)

数学星空

对于\(x^n+y^n+z^n\)的最小值？

当\(n=2k+1\)时

\(a^2 y^{k+1} z^{k+1}+x^{2(k+2)}-y^{2(k+2)}-y^{k+3}z^{k+1}-z^{k+3}y^{k+1}-z^{2(k+2)}=0\)

\(b^2 x^{k+1} z^{k+1}-x^{2(k+2)}+y^{2(k+2)}-x^{k+3}z^{k+1}-z^{k+3}x^{k+1}-z^{2(k+2)}=0\)

\(c^2 x^{k+1} y^{k+1}-x^{2(k+2)}-y^{2(k+2)}-x^{k+3}y^{k+1}-y^{k+3}x^{k+1}+z^{2(k+2)}=0\)

数学星空

对于Fermat-Torricelli Problem 问题：Alexei Yu.Uteshev 在2012.8.16的论文中给出了显式表达。

求\(\triangle P_1 P_2 P_3\) 内一点\(F\),使 \(d={\min}_{(x,y)} F(x_0,y_0)\),三个角分别设为\(\alpha,\beta,\gamma\)

\(F(x,y)=m_1 \sqrt{(x-x_1)^2+(y-y_1)^2}+m_2 \sqrt{(x-x_2)^2+(y-y_2)^2}+m_3 \sqrt{(x-x_3)^2+(y-y_3)^2}\)

即在条件：

\(m_1^2<m_2^2+m_3^2+2m_2 m_3 \cos(\alpha)\)
\(m_2^2<m_1^2+m_3^2+2m_1 m_3 \cos(\beta)\)
\(m_3^2<m_1^2+m_2^2+2m_1 m_2 \cos(\gamma)\)

下，有答案：

\[x_0=\frac{K_1 K_2 K_3}{4S\sigma d}(\frac{x_1}{K_1}+\frac{x_2}{K_2}+\frac{x_3}{K_3})\]

\[y_0=\frac{K_1 K_2 K_3}{4S\sigma d}(\frac{y_1}{K_1}+\frac{y_2}{K_2}+\frac{y_3}{K_3})\]

\[d=\sqrt{d_0}, d_0=\frac{m_1^2 K_1+m_2^2 K_2+m_3^2 K_3}{2\sigma}=2S\sigma+\frac{1}{2}(m_1^2(r_{12}^2+r_{13}^2-r_{23}^2)+m_2^2(r_{23}^2+r_{12}^2-r_{13}^2)+m_3^2(r_{13}^2+r_{23}^2-r_{12}^2))\]

\[r_{jl}=\sqrt{(x_j-x_l)^2+(y_j-y_l)^2}=|P_j P_l| ,\{j,l\}\subset \{1,2,3\}\]

\[S=|x_1 y_2+x_2 y_3+x_3 y_1-x_1 y_3-x_3 y_2-x_2 y_1|\]

\[\sigma=\frac{1}{2}\sqrt{-m_1^4-m_2^4-m_3^4+2m_1^2 m_2^2++2m_2^2 m_3^2++2m_3^2 m_1^2}\]

\[K_1=(r_{12}^2+r_{13}^2-r_{23}^2)\sigma+(m_2^2+m_3^2-m_1^2)S\]

\[K_2=(r_{23}^2+r_{12}^2-r_{13}^2)\sigma+(m_1^2+m_3^2-m_2^2)S\]

\[K_3=(r_{13}^2+r_{23}^2-r_{12}^2)\sigma+(m_1^2+m_2^2-m_3^2)S\]

数学星空

对于\(n=2\),我们可以很容易算出：

\[ \begin{align*}
L_2&=x^2+y^2+z^2=\frac{4}{9}(m_a^2+m_b^2+m_c^2)\\
&=\frac{1}{9}(2a^2+2b^2-c^2+2a^2+2c^2-b^2+2b^2+2c^2-a^2)\\
&=\frac{1}{3}(a^2+b^2+c^2)\\
\end{align*} \]



对于\(n=1\),我们也很容易手工算出

\(x^2+y^2+xy=c^2\)

\(x^2+z^2+xz=b^2\)

\(y^2+z^2+yz=a^2\)

由面积公式易得：

\(\frac{\sqrt{3}}{4}(xy+xz+yz)=S\)

\(L_1^2=(x+y+z)^2=a^2+b^2+c^2-(xy+xz+yz)=\frac{1}{2}(a^2+b^2+c^2)+2\sqrt{3}S\)

且由前面的方程组两两相减，易得：

\((y-z)(x+y+z)=(y-z)L=c^2-b^2\)

\((x-y)(x+y+z)=(x-y)L=b^2-a^2\)

进一步可以得到：

\[x=\frac{1}{3}(L-\frac{2a^2-b^2-c^2}{L})\]

\[y=\frac{1}{3}(L-\frac{2b^2-a^2-c^2}{L})\]

\[z=\frac{1}{3}(L-\frac{2c^2-b^2-a^2}{L})\]

\[L_1=\sqrt{\frac{(a^2+b^2+c^2)+4\sqrt{3}S}{2}}=\sqrt{\frac{(a^2+b^2+c^2)+\sqrt{3(2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4)}}{2}}\]

数学星空

由于对下面成立条件

\[\frac{x^n}{\sin(\alpha)}=\frac{y^n}{\sin(\beta)}=\frac{z^n}{\sin(\gamma)}\]

作代换

\[x \to \frac{1}{x},y \to \frac{1}{y}, z \to \frac{1}{z}\]

即可转换为：

\[\frac{1}{x^n\sin(\alpha)}=\frac{1}{y^n\sin(\beta)}=\frac{1}{z^n\sin(\gamma)}\]

换一种记法：

\[L_{-n}(x,y,z)=L_{n+2} (\frac{1}{x},\frac{1}{y},\frac{1}{z})\]

mathe

用向量法表示$PA,PB,PC$,设它们分别为\(\bar{x},\bar{y},\bar{z}\).
那么对于n对应的问题取极值条件为\(|\bar{x}|^{n-2}\bar{x}+|\bar{y}|^{n-2}\bar{y}+|\bar{z}|^{n-2}\bar{z}=0\)
分别对表达式和$\bar{x},\bar{y}$做向量积就可以得出17#的表达式。
但是另外也可以对表达式分别和$\bar{x},\bar{y}$做内积，可以得出只有余弦的表达式式，然后之际用余弦定理替换角度，得出只包含\(x,y,z,a,b,c\)的表达式

mathe

应该是\(2|\bar{x}|^n+|\bar{y}|^{n-2}(\bar{x}^2+\bar{y}^2-c^2)+|\bar{z}|^{n-2}(\bar{x}^2+\bar{z}^2-b^2)=0\)以及轮换对称