{"id":29253,"date":"2023-09-13T23:35:45","date_gmt":"2023-09-13T16:35:45","guid":{"rendered":"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi"},"modified":"2023-09-13T23:35:45","modified_gmt":"2023-09-13T16:35:45","slug":"trung-diem-la-gi","status":"publish","type":"post","link":"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi","title":{"rendered":"Trung \u0111i\u1ec3m &#8211; Kh\u00e1i ni\u1ec7m v\u00e0 t\u00ednh ch\u1ea5t trung \u0111i\u1ec3m trong h\u00ecnh h\u1ecdc"},"content":{"rendered":"<p>Trung \u0111i\u1ec3m l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng trong h\u1ec7 th\u1ed1ng gi\u00e1o d\u1ee5c \u0111\u1ec3 \u0111\u00e1nh gi\u00e1 hi\u1ec7u su\u1ea5t h\u1ecdc t\u1eadp c\u1ee7a h\u1ecdc sinh. \u0110\u00e2y l\u00e0 m\u1ed9t con s\u1ed1 ho\u1eb7c \u0111i\u1ec3m s\u1ed1 trung b\u00ecnh c\u1ee7a t\u1ea5t c\u1ea3 c\u00e1c b\u00e0i ki\u1ec3m tra, b\u00e0i t\u1eadp v\u00e0 c\u00f4ng vi\u1ec7c kh\u00e1c m\u00e0 h\u1ecdc sinh \u0111\u00e3 ho\u00e0n th\u00e0nh trong m\u1ed9t kho\u1ea3ng th\u1eddi gian nh\u1ea5t \u0111\u1ecbnh. Trung \u0111i\u1ec3m gi\u00fap ph\u1ea3n \u00e1nh s\u1ef1 ph\u00e1t tri\u1ec3n v\u00e0 ti\u1ebfn b\u1ed9 c\u1ee7a h\u1ecdc sinh trong qu\u00e1 tr\u00ecnh h\u1ecdc t\u1eadp.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_40 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" area-label=\"ez-toc-toggle-icon-1\"><label for=\"item-6acfc17e6c92b\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input  type=\"checkbox\" id=\"item-6acfc17e6c92b\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#1_Trung_diem_la_gi\" title=\"1. Trung \u0111i\u1ec3m l\u00e0 g\u00ec?\">1. Trung \u0111i\u1ec3m l\u00e0 g\u00ec?<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#Tinh_chat_cua_trung_diem\" title=\"T\u00ednh ch\u1ea5t c\u1ee7a trung \u0111i\u1ec3m:\">T\u00ednh ch\u1ea5t c\u1ee7a trung \u0111i\u1ec3m:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#2_Tinh_chat_trung_diem_cua_doan_thang\" title=\"2. T\u00ednh ch\u1ea5t trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng\">2. T\u00ednh ch\u1ea5t trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#Vi_du\" title=\"V\u00ed d\u1ee5:\">V\u00ed d\u1ee5:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#3_Cach_ve_trung_diem_cua_doan_thang\" title=\"3. C\u00e1ch v\u1ebd trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng\">3. C\u00e1ch v\u1ebd trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#31_Dinh_nghia_trung_diem\" title=\"3.1. \u0110\u1ecbnh ngh\u0129a trung \u0111i\u1ec3m\">3.1. \u0110\u1ecbnh ngh\u0129a trung \u0111i\u1ec3m<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#32_Cach_ve_trung_diem\" title=\"3.2. C\u00e1ch v\u1ebd trung \u0111i\u1ec3m\">3.2. C\u00e1ch v\u1ebd trung \u0111i\u1ec3m<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#4_Cach_chung_minh_trung_diem_theo_dinh_nghia\" title=\"4. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m theo \u0111\u1ecbnh ngh\u0129a\">4. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m theo \u0111\u1ecbnh ngh\u0129a<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#5_Cach_chung_minh_trung_diem_trong_tam_giac\" title=\"5. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong tam gi\u00e1c\">5. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong tam gi\u00e1c<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#51_Dinh_ly_ve_trung_diem_trong_tam_giac\" title=\"5.1. \u0110\u1ecbnh l\u00fd v\u1ec1 trung \u0111i\u1ec3m trong tam gi\u00e1c\">5.1. \u0110\u1ecbnh l\u00fd v\u1ec1 trung \u0111i\u1ec3m trong tam gi\u00e1c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#52_Cach_chung_minh_trung_diem_trong_tam_giac\" title=\"5.2. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong tam gi\u00e1c\">5.2. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong tam gi\u00e1c<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#6_Cach_chung_minh_trung_diem_trong_tu_giac\" title=\"6. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong t\u1ee9 gi\u00e1c\">6. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong t\u1ee9 gi\u00e1c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#7_Cach_chung_minh_trung_diem_lien_quan_den_duong_tron\" title=\"7. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m li\u00ean quan \u0111\u1ebfn \u0111\u01b0\u1eddng tr\u00f2n\">7. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m li\u00ean quan \u0111\u1ebfn \u0111\u01b0\u1eddng tr\u00f2n<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#71_Dinh_ly_ve_trung_diem_va_duong_kinh\" title=\"7.1. \u0110\u1ecbnh l\u00fd v\u1ec1 trung \u0111i\u1ec3m v\u00e0 \u0111\u01b0\u1eddng k\u00ednh\">7.1. \u0110\u1ecbnh l\u00fd v\u1ec1 trung \u0111i\u1ec3m v\u00e0 \u0111\u01b0\u1eddng k\u00ednh<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/ungdung.hotronghiencuu.com\/open\/trung-diem-la-gi\/#72_Cach_chung_minh_trung_diem_lien_quan_den_duong_tron\" title=\"7.2. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m li\u00ean quan \u0111\u1ebfn \u0111\u01b0\u1eddng tr\u00f2n\">7.2. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m li\u00ean quan \u0111\u1ebfn \u0111\u01b0\u1eddng tr\u00f2n<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"1_Trung_diem_la_gi\"><\/span>1. Trung \u0111i\u1ec3m l\u00e0 g\u00ec?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Trung \u0111i\u1ec3m l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m trong h\u00ecnh h\u1ecdc, n\u00f3 \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh nh\u01b0 m\u1ed9t \u0111i\u1ec3m n\u1eb1m \u1edf gi\u1eefa c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng, c\u00e1ch \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n th\u1eb3ng m\u1ed9t kho\u1ea3ng b\u1eb1ng nhau. N\u1ebfu ch\u00fang ta k\u00fd hi\u1ec7u c\u00e1c \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n th\u1eb3ng l\u00e0 A v\u00e0 B, th\u00ec trung \u0111i\u1ec3m c\u1ee7a AB s\u1ebd \u0111\u01b0\u1ee3c k\u00fd hi\u1ec7u l\u00e0 M. \u0110i\u1ec1u quan tr\u1ecdng \u0111\u1ec3 l\u01b0u \u00fd l\u00e0 trung \u0111i\u1ec3m kh\u00f4ng ch\u1ec9 \u00e1p d\u1ee5ng cho c\u00e1c \u0111o\u1ea1n th\u1eb3ng hai chi\u1ec1u m\u00e0 c\u00f2n c\u1ea3 c\u00e1c c\u1ea1nh trong tam gi\u00e1c v\u00e0 t\u1ee9 gi\u00e1c.<\/p>\n<p>Trong h\u00ecnh h\u1ecdc Euclid, trung \u0111i\u1ec3m \u0111\u01b0\u1ee3c coi l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n v\u00e0 c\u00f3 vai tr\u00f2 quan tr\u1ecdng trong vi\u1ec7c x\u00e2y d\u1ef1ng c\u00e1c ph\u00e9p bi\u1ebfn h\u00ecnh v\u00e0 ch\u1ee9ng minh c\u00e1c t\u00ednh ch\u1ea5t c\u1ee7a c\u00e1c h\u00ecnh h\u1ecdc.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tinh_chat_cua_trung_diem\"><\/span>T\u00ednh ch\u1ea5t c\u1ee7a trung \u0111i\u1ec3m:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li>Trung \u0111i\u1ec3m chia \u0111o\u1ea1n th\u1eb3ng th\u00e0nh hai ph\u1ea7n c\u00f3 c\u00f9ng \u0111\u1ed9 d\u00e0i.<\/li>\n<li>Hai tia t\u1eeb trung \u0111i\u1ec3m t\u1edbi hai \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n th\u1eb3ng l\u00e0 hai tia c\u00f9ng h\u01b0\u1edbng v\u00e0 c\u00f9ng chi\u1ec1u d\u00e0i.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"2_Tinh_chat_trung_diem_cua_doan_thang\"><\/span>2. T\u00ednh ch\u1ea5t trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>T\u00ednh ch\u1ea5t quan tr\u1ecdng nh\u1ea5t c\u1ee7a trung \u0111i\u1ec3m l\u00e0 n\u00f3 chia \u0111o\u1ea1n th\u1eb3ng th\u00e0nh hai ph\u1ea7n c\u00f3 c\u00f9ng \u0111\u1ed9 d\u00e0i. \u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb m\u1ed9t \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n th\u1eb3ng t\u1edbi trung \u0111i\u1ec3m s\u1ebd b\u1eb1ng kho\u1ea3ng c\u00e1ch t\u1eeb trung \u0111i\u1ec3m t\u1edbi \u0111\u1ea7u m\u00fat c\u00f2n l\u1ea1i.<\/p>\n<p>Ngo\u00e0i ra, hai tia t\u1eeb trung \u0111i\u1ec3m t\u1edbi hai \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n th\u1eb3ng l\u00e0 hai tia c\u00f9ng h\u01b0\u1edbng v\u00e0 c\u00f3 chi\u1ec1u d\u00e0i b\u1eb1ng nhau. \u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 khi ta k\u00e9o c\u00e1c tia n\u00e0y ra xa, ch\u00fang s\u1ebd ti\u1ebfp t\u1ee5c song song v\u00e0 kh\u00f4ng bao gi\u1edd giao nhau.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Vi_du\"><\/span>V\u00ed d\u1ee5:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Gi\u1ea3 s\u1eed ta c\u00f3 m\u1ed9t \u0111o\u1ea1n th\u1eb3ng AB v\u1edbi A(1, 2) v\u00e0 B(5, 6). \u0110\u1ec3 t\u00ecm trung \u0111i\u1ec3m M c\u1ee7a AB, ta s\u1eed d\u1ee5ng c\u00f4ng th\u1ee9c sau: <\/p>\n<p>M = ((x1 + x2) \/ 2, (y1 + y2) \/ 2)<\/p>\n<p>\u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c n\u00e0y v\u00e0o v\u00ed d\u1ee5 tr\u00ean, ta t\u00ednh \u0111\u01b0\u1ee3c r\u1eb1ng M = ((1 + 5) \/ 2, (2 + 6) \/ 2) = (3, 4). V\u1eady trung \u0111i\u1ec3m M c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB l\u00e0 (3, 4).<\/p>\n<h2><span class=\"ez-toc-section\" id=\"3_Cach_ve_trung_diem_cua_doan_thang\"><\/span>3. C\u00e1ch v\u1ebd trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"31_Dinh_nghia_trung_diem\"><\/span>3.1. \u0110\u1ecbnh ngh\u0129a trung \u0111i\u1ec3m<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Trong h\u00ecnh h\u1ecdc, trung \u0111i\u1ec3m c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng l\u00e0 \u0111i\u1ec3m n\u1eb1m \u1edf gi\u1eefa \u0111o\u1ea1n th\u1eb3ng \u0111\u00f3 v\u00e0 c\u00f3 c\u00f9ng kho\u1ea3ng c\u00e1ch t\u1edbi hai \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n th\u1eb3ng.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"32_Cach_ve_trung_diem\"><\/span>3.2. C\u00e1ch v\u1ebd trung \u0111i\u1ec3m<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>C\u00f3 m\u1ed9t ph\u01b0\u01a1ng ph\u00e1p r\u1ea5t \u0111\u01a1n gi\u1ea3n \u0111\u1ec3 v\u1ebd trung \u0111i\u1ec3m c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng. Ta ch\u1ec9 c\u1ea7n s\u1eed d\u1ee5ng b\u00fat v\u1ebd v\u00e0 th\u01b0\u1edbc \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh hai \u0111i\u1ec3m cu\u1ed1i c\u1ee7a \u0111o\u1ea1n th\u1eb3ng, sau \u0111\u00f3 d\u00f9ng th\u01b0\u1edbc \u0111\u1ec3 n\u1ed1i hai \u0111i\u1ec3m n\u00e0y l\u1ea1i v\u00e0 chia ph\u1ea7n c\u00f2n l\u1ea1i th\u00e0nh hai ph\u1ea7n b\u1eb1ng nhau. \u0110i\u1ec3m ch\u00ednh gi\u1eefa l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"4_Cach_chung_minh_trung_diem_theo_dinh_nghia\"><\/span>4. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m theo \u0111\u1ecbnh ngh\u0129a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><img decoding=\"async\" alt=\"4. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m theo \u0111\u1ecbnh ngh\u0129a\" src=\"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-content\/uploads\/2023\/09\/trung_diem_khai_niem_va_tinh_chat_trung_diem_trong_hinh_hoc.jpeg\" style=\"text-align: center; clear: both; display: block; margin: 0 auto;\" title=\"4. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m theo \u0111\u1ecbnh ngh\u0129a\"\/><\/p>\n<p>\u0110\u1ec3 ch\u1ee9ng minh r\u1eb1ng m\u1ed9t \u0111i\u1ec3m l\u00e0 trung \u0111i\u1ec3m c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng theo \u0111\u1ecbnh ngh\u0129a, ta c\u1ea7n ch\u1ee9ng minh r\u1eb1ng kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m n\u00e0y t\u1edbi hai \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n th\u1eb3ng l\u00e0 b\u1eb1ng nhau.<\/p>\n<p>Gi\u1ea3 s\u1eed ta c\u00f3 m\u1ed9t \u0111o\u1ea1n th\u1eb3ng AB v\u00e0 m\u1ed9t \u0111i\u1ec3m M n\u1eb1m gi\u1eefa hai \u0111\u1ea7u m\u00fat A v\u00e0 B. \u0110\u1ec3 ch\u1ee9ng minh r\u1eb1ng M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, ta c\u1ea7n ch\u1ee9ng minh AM = MB. Ta c\u00f3 th\u1ec3 s\u1eed d\u1ee5ng c\u00e1c ph\u01b0\u01a1ng ph\u00e1p ch\u1ee9ng minh h\u00ecnh h\u1ecdc nh\u01b0 s\u1eed d\u1ee5ng c\u00f4ng th\u1ee9c kho\u1ea3ng c\u00e1ch ho\u1eb7c s\u1eed d\u1ee5ng c\u00e1c \u0111\u1ecbnh l\u00fd li\u00ean quan \u0111\u1ec3 ch\u1ee9ng minh \u0111i\u1ec1u n\u00e0y.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"5_Cach_chung_minh_trung_diem_trong_tam_giac\"><\/span>5. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong tam gi\u00e1c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"51_Dinh_ly_ve_trung_diem_trong_tam_giac\"><\/span>5.1. \u0110\u1ecbnh l\u00fd v\u1ec1 trung \u0111i\u1ec3m trong tam gi\u00e1c<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Trong tam gi\u00e1c ABC, M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng BC n\u1ebfu v\u00e0 ch\u1ec9 n\u1ebfu AM l\u00e0 \u0111\u01b0\u1eddng cao c\u1ee7a tam gi\u00e1c ABC.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"52_Cach_chung_minh_trung_diem_trong_tam_giac\"><\/span>5.2. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong tam gi\u00e1c<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0110\u1ec3 ch\u1ee9ng minh r\u1eb1ng m\u1ed9t \u0111i\u1ec3m l\u00e0 trung \u0111i\u1ec3m c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng trong tam gi\u00e1c, ta c\u00f3 th\u1ec3 s\u1eed d\u1ee5ng c\u00e1c ph\u01b0\u01a1ng ph\u00e1p ch\u1ee9ng minh h\u00ecnh h\u1ecdc nh\u01b0 s\u1eed d\u1ee5ng c\u00e1c \u0111\u1ecbnh l\u00fd v\u1ec1 t\u1ec9 l\u1ec7, g\u00f3c hay c\u1ea1nh t\u01b0\u01a1ng t\u1ef1 trong tam gi\u00e1c \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh v\u1ecb tr\u00ed c\u1ee7a trung \u0111i\u1ec3m.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"6_Cach_chung_minh_trung_diem_trong_tu_giac\"><\/span>6. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong t\u1ee9 gi\u00e1c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><img decoding=\"async\" alt=\"6. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong t\u1ee9 gi\u00e1c\" src=\"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-content\/uploads\/2023\/09\/trung_diem_khai_niem_va_tinh_chat_trung_diem_trong_hinh_hoc-1.jpeg\" style=\"text-align: center; clear: both; display: block; margin: 0 auto;\" title=\"6. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m trong t\u1ee9 gi\u00e1c\"\/><\/p>\n<p>Trong t\u1ee9 gi\u00e1c ABCD, M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB n\u1ebfu v\u00e0 ch\u1ec9 n\u1ebfu AM = MB v\u00e0 MC = MD.<\/p>\n<p>\u0110\u1ec3 ch\u1ee9ng minh r\u1eb1ng m\u1ed9t \u0111i\u1ec3m l\u00e0 trung \u0111i\u1ec3m c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng trong t\u1ee9 gi\u00e1c, ta c\u1ea7n ch\u1ee9ng minh r\u1eb1ng hai c\u1eb7p \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 c\u00f9ng \u0111\u1ed9 d\u00e0i. Ta c\u00f3 th\u1ec3 s\u1eed d\u1ee5ng c\u00e1c ph\u01b0\u01a1ng ph\u00e1p ch\u1ee9ng minh h\u00ecnh h\u1ecdc nh\u01b0 s\u1eed d\u1ee5ng c\u00e1c \u0111\u1ecbnh l\u00fd v\u1ec1 t\u1ec9 l\u1ec7 ho\u1eb7c s\u1eed d\u1ee5ng c\u00e1c c\u00f4ng th\u1ee9c kho\u1ea3ng c\u00e1ch \u0111\u1ec3 ch\u1ee9ng minh \u0111i\u1ec1u n\u00e0y.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"7_Cach_chung_minh_trung_diem_lien_quan_den_duong_tron\"><\/span>7. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m li\u00ean quan \u0111\u1ebfn \u0111\u01b0\u1eddng tr\u00f2n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"71_Dinh_ly_ve_trung_diem_va_duong_kinh\"><\/span>7.1. \u0110\u1ecbnh l\u00fd v\u1ec1 trung \u0111i\u1ec3m v\u00e0 \u0111\u01b0\u1eddng k\u00ednh<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n, m\u1ed9t \u0111o\u1ea1n th\u1eb3ng c\u00f3 m\u1ed9t \u0111i\u1ec3m l\u00e0 trung \u0111i\u1ec3m c\u1ee7a n\u00f3 n\u1ebfu v\u00e0 ch\u1ec9 n\u1ebfu n\u00f3 l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"72_Cach_chung_minh_trung_diem_lien_quan_den_duong_tron\"><\/span>7.2. C\u00e1ch ch\u1ee9ng minh trung \u0111i\u1ec3m li\u00ean quan \u0111\u1ebfn \u0111\u01b0\u1eddng tr\u00f2n<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0110\u1ec3 ch\u1ee9ng minh r\u1eb1ng m\u1ed9t \u0111i\u1ec3m l\u00e0 trung \u0111i\u1ec3m c\u1ee7a m\u1ed9t \u0111o\u1ea1n th\u1eb3ng li\u00ean quan \u0111\u1ebfn \u0111\u01b0\u1eddng tr\u00f2n, ta c\u00f3 th\u1ec3 s\u1eed d\u1ee5ng c\u00e1c ph\u01b0\u01a1ng ph\u00e1p ch\u1ee9ng minh h\u00ecnh h\u1ecdc nh\u01b0 s\u1eed d\u1ee5ng c\u00e1c \u0111\u1ecbnh l\u00fd v\u1ec1 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n, g\u00f3c n\u1ed9i ti\u1ebfp hay cung t\u01b0\u01a1ng \u1ee9ng \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh v\u1ecb tr\u00ed c\u1ee7a trung \u0111i\u1ec3m.<\/p>\n<p>Trung \u0111i\u1ec3m l\u00e0 m\u1ed9t h\u1ec7 th\u1ed1ng \u0111\u00e1nh gi\u00e1 th\u00e0nh t\u00edch h\u1ecdc t\u1eadp c\u1ee7a h\u1ecdc sinh d\u1ef1a tr\u00ean c\u00e1c b\u00e0i ki\u1ec3m tra v\u00e0 c\u00f4ng vi\u1ec7c trong qu\u00e1 tr\u00ecnh h\u1ecdc. \u0110i\u1ec3m trung \u0111i\u1ec3m c\u00f3 vai tr\u00f2 quan tr\u1ecdng trong vi\u1ec7c x\u00e1c \u0111\u1ecbnh n\u0103ng l\u1ef1c v\u00e0 ti\u1ebfn b\u1ed9 c\u1ee7a h\u1ecdc sinh, gi\u00fap ph\u1ee5 huynh v\u00e0 gi\u00e1o vi\u00ean hi\u1ec3u r\u00f5 v\u1ec1 s\u1ef1 ph\u00e1t tri\u1ec3n c\u1ee7a con em m\u00ecnh. Qua \u0111\u00f3, trung \u0111i\u1ec3m g\u00f3p ph\u1ea7n t\u1ea1o \u0111\u1ed9ng l\u1ef1c cho h\u1ecdc sinh n\u1ed7 l\u1ef1c r\u00e8n luy\u1ec7n v\u00e0 ho\u00e0n thi\u1ec7n k\u1ef9 n\u0103ng.<br \/>\n<iframe loading=\"lazy\" title=\"Trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng - To\u00e1n h\u1ecdc 6 - C\u00f4 Nguy\u1ec5n Di\u1ec7u Linh ( D\u1ec4 HI\u1ec2U NH\u1ea4T )\" width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/ZxwvJqU1-dY?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Trung \u0111i\u1ec3m l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng trong h\u1ec7 th\u1ed1ng gi\u00e1o d\u1ee5c \u0111\u1ec3 \u0111\u00e1nh gi\u00e1 hi\u1ec7u su\u1ea5t h\u1ecdc t\u1eadp c\u1ee7a h\u1ecdc sinh. \u0110\u00e2y l\u00e0 m\u1ed9t&#8230; <\/p>\n","protected":false},"author":1,"featured_media":29251,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-29253","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-khai-niem-1"],"_links":{"self":[{"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/posts\/29253","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/comments?post=29253"}],"version-history":[{"count":0,"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/posts\/29253\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/media\/29251"}],"wp:attachment":[{"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/media?parent=29253"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/categories?post=29253"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ungdung.hotronghiencuu.com\/open\/wp-json\/wp\/v2\/tags?post=29253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}