Ghi ch\u00fa: T\u00e0i li\u1ec7u tr\u00edch d\u1eabn<\/p>\n\n\n\n
B\u1ed9 m\u00f4n to\u00e1n h\u00ecnh h\u1ecdc lu\u00f4n mang l\u1ea1i cho ch\u00fang ta c\u1ea3m gi\u00e1c c\u1ef1c k\u1ef3 th\u00fa v\u1ecb. Tuy nhi\u00ean vi\u1ec7c ghi nh\u1edb c\u00e1c c\u00f4ng th\u1ee9c hay ph\u00e2n bi\u1ec7t t\u00ednh ch\u1ea5t c\u0169ng khi\u1ebfn nhi\u1ec1u h\u1ecdc sinh<\/a> ng\u00e1n ng\u1ea9m. \u0110\u1eebng qu\u00e1 lo l\u1eafng nh\u00e9! Vietlearn s\u1ebd \u0111\u1ed3ng h\u00e0nh c\u00f9ng b\u1ea1n trong t\u1eebng b\u00e0i h\u1ecdc. H\u00f4m nay, ch\u00fang ta h\u00e3y c\u00f9ng \u00f4n t\u1eadp c\u00e1c ki\u1ebfn th\u1ee9c quan tr\u1ecdng c\u1ee7a t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c th\u00f4i n\u00e0o!<\/p>\n\n\n\n \u0110\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c l\u00e0 g\u00ec?<\/p>\n\n\n\n T\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c l\u00e0 g\u00ec?<\/p>\n\n\n\n \u0110\u1ecbnh ngh\u0129a v\u1ec1 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c \u0111\u01b0\u1ee3c ph\u00e1t bi\u1ec3u nh\u01b0 sau: \u201cTrong m\u1ed9t tam gi\u00e1c, \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a m\u1ed7i c\u1ea1nh g\u1ecdi l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c \u0111\u00f3.\u201d<\/p>\n\n\n\n Ch\u1eb3ng h\u1ea1n nh\u01b0 trong tam gi\u00e1c ABC: a l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c \u1ee9ng v\u1edbi c\u1ea1nh BC, b l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c \u1ee9ng v\u1edbi c\u1ea1nh AC v\u00e0 c l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c \u1ee9ng v\u1edbi c\u1ea1nh AB.<\/p>\n\n\n\n Trong m\u1ed7i tam gi\u00e1c \u0111\u1ec1u c\u00f3 ba \u0111\u01b0\u1eddng trung tr\u1ef1c.<\/p>\n\n\n\n T\u00ednh ch\u1ea5t c\u1ee7a \u0111\u01b0\u1eddng trung tr\u1ef1c: Trong m\u1ed9t tam gi\u00e1c c\u00e2n, \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a c\u1ea1nh \u0111\u00e1y \u0111\u1ed3ng th\u1eddi l\u00e0 \u0111\u01b0\u1eddng trung tuy\u1ebfn \u1ee9ng v\u1edbi c\u1ea1nh n\u00e0y.<\/p>\n\n\n\n T\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c<\/p>\n\n\n\n T\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c c\u1ee5 th\u1ec3 nh\u01b0 sau:<\/p>\n\n\n\n B\u1ea5t k\u1ef3 tam gi\u00e1c n\u00e0o c\u0169ng \u0111\u01b0\u1ee3c s\u1edf h\u1eefu 3 \u0111\u01b0\u1eddng trung tr\u1ef1c, t\u00ednh ch\u1ea5t chung c\u1ee7a 3 \u0111\u01b0\u1eddng n\u00e0y nh\u01b0 sau: C\u00f9ng \u0111i qua m\u1ed9t \u0111i\u1ec3m, \u0111i\u1ec3m n\u00e0y c\u00e1ch \u0111\u1ec1u ba \u0111\u1ec9nh c\u1ee7a tam gi\u00e1c \u0111\u00f3.<\/p>\n\n\n\n Ch\u1eb3ng h\u1ea1n nh\u01b0: O l\u00e0 giao \u0111i\u1ec3m ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c ABC, suy ra ta c\u00f3 OA = OB = OC<\/p>\n\n\n\n L\u01b0u \u00fd: n\u1ebfu m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n l\u1ea5y giao \u0111i\u1ec3m c\u1ee7a 3 \u0111\u01b0\u1eddng trung tr\u1ef1c l\u00e0m t\u00e2m v\u00e0 \u0111i qua 3 \u0111\u1ec9nh c\u1ee7a tam gi\u00e1c, th\u00ec \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00f3 \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c.<\/p>\n\n\n\n Ch\u1eb3ng h\u1ea1n nh\u01b0: O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a 3 \u0111\u01b0\u1eddng trung tr\u1ef1c tam gi\u00e1c ABC v\u00e0 m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111i qua ba \u0111\u1ec9nh A, B, C; th\u00ec \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00f3 l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC.<\/p>\n\n\n\n \u0110\u1ec3 hi\u1ec3u r\u00f5 h\u01a1n v\u1ec1 t\u00ednh ch\u1ea5t \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a m\u1ed9t tam gi\u00e1c, c\u00f3 th\u1ec3 t\u00ecm \u0111\u1ecdc th\u00eam 1 s\u1ed1 b\u00e0i vi\u1ebft kh\u00e1c c\u1ee7a Vietlearn.<\/p>\n\n\n\n T\u00ednh ch\u1ea5t \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng<\/p><\/blockquote>\n\n\n\n M\u1ed9t s\u1ed1 b\u00e0i t\u1eadp tr\u1eafc nghi\u1ec7m \u1ee9ng d\u1ee5ng t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c<\/p>\n\n\n\n C\u1ee7ng c\u1ed1 l\u00fd thuy\u1ebft \u0111\u00e3 h\u1ecdc qua c\u00e1c b\u00e0i t\u1eadp<\/p>\n\n\n\n B\u00e0i t\u1eadp 1<\/p>\n\n\n\n Cho \u0394ABC c\u00f3 hai \u0111\u01b0\u1eddng cao BD v\u00e0 CE, g\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. Em h\u00e3y ch\u1ecdn c\u00e2u sai:<\/p>\n\n\n\n BM = MC<\/p>\n\n\n\n ME = MD<\/p>\n\n\n\n DM = MB<\/p>\n\n\n\n M kh\u00f4ng thu\u1ed9c \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a c\u1ea1nh DE<\/p>\n\n\n\n Ta c\u00f3: M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC, suy ra theo t\u00ednh ch\u1ea5t trung \u0111i\u1ec3m th\u00ec BM = MC,lo\u1ea1i \u0111\u00e1p \u00e1n A.<\/p>\n\n\n\n X\u00e9t \u0394BCE c\u00f3 M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. Suy ra EM ch\u00ednh l\u00e0 trung tuy\u1ebfn<\/p>\n\n\n\n Ta c\u00f3 l\u00fd thuy\u1ebft: Trong tam gi\u00e1c vu\u00f4ng, \u0111\u01b0\u1eddng trung tuy\u1ebfn \u1ee9ng v\u1edbi c\u1ea1nh huy\u1ec1n b\u1eb1ng n\u1eeda c\u1ea1nh \u0111\u00f3.\u21d2EM = BC\/2 (1)<\/p>\n\n\n\n Ti\u1ebfp t\u1ee5c x\u00e9t \u0394BCD c\u00f3 M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. Suy ra DM c\u0169ng l\u00e0 trung tuy\u1ebfn<\/p>\n\n\n\n \u21d2 DM = MB = BC\/2 (2), n\u00ean lo\u1ea1i \u0111\u00e1p \u00e1n C<\/p>\n\n\n\n T\u1eeb (1) v\u00e0 (2) suy ra: EM = DM \u21d2 M thu\u1ed9c \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a DE, lo\u1ea1i \u0111\u01b0\u1ee3c \u0111\u00e1p \u00e1n D, ch\u1ecdn \u0111\u00e1p \u00e1n B.<\/p>\n\n\n\n B\u00e0i t\u1eadp 2<\/p>\n\n\n\n Cho \u0394ABC c\u00f3 AC > AB, t\u1ea1i AC l\u1ea5y \u0111i\u1ec3m E sao cho CE = AB, O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a c\u00e1c \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a BE v\u00e0 AC. Ch\u1ecdn \u0111\u00e1p \u00e1n \u0111\u00fang:<\/p>\n\n\n\n \u0394ABO = \u0394COE<\/p>\n\n\n\n \u0394BOA = \u0394COE<\/p>\n\n\n\n \u0394AOB = \u0394COE<\/p>\n\n\n\n \u0394ABO = \u0394CEO<\/p>\n\n\n\n X\u00e9t tam gi\u00e1c \u0394AOB v\u00e0 \u0394COE\u201d<\/p>\n\n\n\n O thu\u1ed9c \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a AC\u21d2 OA = OC<\/p>\n\n\n\n O thu\u1ed9c \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a BE\u21d2 OB = OE<\/p>\n\n\n\n Theo gi\u1ea3 thi\u1ebft: AB = CE<\/p>\n\n\n\n Do \u0111\u00f3 \u0394AOB = \u0394COE (c\u1ea1nh-c\u1ea1nh-c\u1ea1nh)<\/p>\n\n\n\n Ch\u1ecdn \u0111\u00e1p \u00e1n C<\/p>\n\n\n\n B\u00e0i t\u1eadp 3<\/p>\n\n\n\n Cho \u0394ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH, t\u1ea1i c\u1ea1nh AC l\u1ea5y \u0111i\u1ec3m K sao cho AK = AH, KD \u22a5 AC (D \u2208 BC). Ch\u1ecdn c\u00e2u \u0111\u00fang<\/p>\n\n\n\n \u0394AHD = \u0394AKD<\/p>\n\n\n\n AD l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a HK<\/p>\n\n\n\n AD l\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c HAK<\/p>\n\n\n\n C\u1ea3 A, B, C \u0111\u1ec1u \u0111\u00fang<\/p>\n\n\n\n X\u00e9t tam gi\u00e1c vu\u00f4ng AHD v\u00e0 AKD c\u00f3:<\/p>\n\n\n\n AH = AK (gi\u1ea3 thi\u1ebft)<\/p>\n\n\n\n AD chung<\/p>\n\n\n\n Suy ra \u0394AHD = \u0394AKD (c\u1ea1nh huy\u1ec1n-c\u1ea1nh g\u00f3c vu\u00f4ng) n\u00ean c\u00e2u A \u0111\u00fang<\/p>\n\n\n\n Ta c\u00f3: HD = DK; \u2220HAD = \u2220DAK.<\/p>\n\n\n\n Suy ra AD l\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c HAK, n\u00ean c\u00e2u C \u0111\u00fang<\/p>\n\n\n\n Ta l\u1ea1i c\u00f3: AH = AK (g\u1ec9a thi\u1ebft) v\u00e0 HA = DK (cmt). Suy ra AD l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng HK n\u00ean B \u0111\u00fang.<\/p>\n\n\n\n V\u1eady A, B, C \u0111\u1ec1u \u0111\u00fang. Ch\u1ecdn \u0111\u00e1p \u00e1n D<\/p>\n\n\n\n M\u1ed9t s\u1ed1 b\u00e0i t\u1eadp t\u1ef1 lu\u1eadn \u1ee9ng d\u1ee5ng t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c<\/p>\n\n\n\n \u00c1p d\u1ee5ng l\u00fd thuy\u1ebft gi\u1ea3i c\u00e1c b\u00e0i t\u1eadp t\u1ef1 lu\u1eadn<\/p>\n\n\n\n B\u00e0i t\u1eadp 1<\/p>\n\n\n\n Cho tam gi\u00e1c ABC, AK l\u00e0 \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c A, giao \u0111i\u1ec3m \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a tam gi\u00e1c ABK tr\u00f9ng v\u1edbi giao \u0111i\u1ec3m ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c ABC. Y\u00eau c\u1ea7u: T\u00ednh s\u1ed1 \u0111o c\u00e1c g\u00f3c c\u1ee7a tam gi\u00e1c ABC.<\/p>\n\n\n\n B\u00e0i gi\u1ea3i:<\/p>\n\n\n\n G\u1ecdi O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a 3 \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a tam gi\u00e1c ABC, O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c ABC (gi\u1ea3 thi\u1ebft)<\/p>\n\n\n\n Suy ra:<\/p>\n\n\n\n OA = OB = OC<\/p>\n\n\n\n C\u00e1c tam gi\u00e1c AOB, AOC, BOC l\u00e0 c\u00e1c tam gi\u00e1c \u0111\u1ec1u.<\/p>\n\n\n\n AK l\u00e0 \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c BAC (gi\u1ea3 thi\u1ebft). Suy ra: n\u1ebfu \u2220KAB = 2x th\u00ec \u2220BAC = 4x<\/p>\n\n\n\n Ta c\u00f3: \u0394AOB = \u0394COB. Suy ra: AB = CB<\/p>\n\n\n\n V\u1eady tam gi\u00e1c ABC c\u00e2n t\u1ea1i \u0111\u1ec9nh B<\/p>\n\n\n\n \u21d2 \u2220BAC = \u2220BCA<\/p>\n\n\n\n Khi \u0111\u00f3 ta c\u00f3:<\/p>\n\n\n\n 2x + 4x + 4x = 180\u00b0 \u21d2 10x = 180\u00b0 \u21d2 x =18\u00b0<\/p>\n\n\n\n V\u1eady c\u00f3 th\u1ec3 k\u1ebft lu\u1eadn s\u1ed1 \u0111o c\u00e1c g\u00f3c c\u1ee7a tam gi\u00e1c ABC l\u00e0: \u2220A = \u2220C = 72\u00b0, \u2220B = 18\u00b0<\/p>\n\n\n\n B\u00e0i t\u1eadp 2<\/p>\n\n\n\n Cho tam gi\u00e1c \u0111\u1ec1u ABC, t\u1ea1i ba c\u1ea1nh AB, BC v\u00e0 CA l\u1ea5y c\u00e1c \u0111i\u1ec3m theo th\u1ee9 t\u1ef1 M, N, P sao cho AM = BN = CP., O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a ba \u0111\u01b0\u1eddng trung tr\u1ef1c. Y\u00eau c\u1ea7u: Ch\u1ee9ng minh O c\u0169ng l\u00e0 giao \u0111i\u1ec3m ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c MNP.<\/p>\n\n\n\n B\u00e0i gi\u1ea3i:<\/p>\n\n\n\n O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c ABC( gi\u1ea3 thi\u1ebft). Suy ra: OA = OB = OC\u21d2 C\u00e1c tam gi\u00e1c AOM, BON, COP c\u00f3:<\/p>\n\n\n\n AM = BN = CP (gt)<\/p>\n\n\n\n Do \u0111\u00f3: \u0394AOM = \u0394BON = \u0394COP (c\u1ea1nh-g\u00f3c-c\u1ea1nh)<\/p>\n\n\n\n \u21d2 OM = ON = OP<\/p>\n\n\n\n Hay n\u00f3i c\u00e1ch kh\u00e1c: O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a ba \u0111\u01b0\u1eddng trung tr\u1ef1c tam gi\u00e1c MNP<\/p>\n\n\n\n Tr\u00ean \u0111\u00e2y l\u00e0 m\u1ed9t s\u1ed1 ki\u1ebfn th\u1ee9c l\u00fd thuy\u1ebft v\u00e0 b\u00e0i t\u1eadp v\u1ec1 t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a tam gi\u00e1c m\u00e0 ch\u00fang t\u00f4i mu\u1ed1n chia s\u1ebb \u0111\u1ebfn c\u00e1c b\u1ea1n. Hy v\u1ecdng c\u00e1c b\u1ea1n \u0111\u00e3 c\u00f3 nh\u1eefng ph\u00fat gi\u00e2y h\u1ecdc t\u1eadp<\/a> th\u1eadt b\u1ed5 \u00edch v\u1edbi Vietlearn!<\/p>\n\n\n\n Xem ngay:<\/p>\n\n\n\n \u0110\u01b0\u1eddng trung b\u00ecnh c\u1ee7a h\u00ecnh thang v\u00e0 c\u00e1c d\u1ea1ng b\u00e0i t\u1eadp<\/p>\n\n\n\n\n\n\n","protected":false},"excerpt":{"rendered":" Ghi ch\u00fa: T\u00e0i li\u1ec7u tr\u00edch d\u1eabn B\u1ed9 m\u00f4n to\u00e1n h\u00ecnh h\u1ecdc lu\u00f4n mang l\u1ea1i cho ch\u00fang ta c\u1ea3m gi\u00e1c c\u1ef1c k\u1ef3 th\u00fa v\u1ecb. Tuy nhi\u00ean vi\u1ec7c ghi nh\u1edb c\u00e1c c\u00f4ng th\u1ee9c hay ph\u00e2n bi\u1ec7t t\u00ednh ch\u1ea5t c\u0169ng khi\u1ebfn nhi\u1ec1u h\u1ecdc sinh ng\u00e1n ng\u1ea9m. \u0110\u1eebng qu\u00e1 lo l\u1eafng nh\u00e9! Vietlearn s\u1ebd \u0111\u1ed3ng h\u00e0nh c\u00f9ng b\u1ea1n trong […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_disable_autopaging":false},"categories":[9],"tags":[],"class_list":["post-969","post","type-post","status-publish","format-standard","hentry","category-toan-hoc"],"_links":{"self":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/969","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/comments?post=969"}],"version-history":[{"count":0,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/969\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/media?parent=969"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/categories?post=969"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/tags?post=969"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}