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*Phân tíchGiả sử đã dựng được điểm Acó D nằm trên trung trực QBcó DQ =DP (vì Q đối xứng P qua DE) =EA (1)Có: EDQ^=EDP^=DAE^" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">ˆEDQ=ˆEDP=ˆDAEEDQ^=EDP^=DAE^ (2)từ (1, 2) ⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ AEDQ là hình thang cân⇒AQ//DE" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒AQ//DE⇒AQ//DE , mà DE⊥" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⊥⊥PQ⇒AQ⊥QP" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒AQ⊥QP⇒AQ⊥QP *Thay đổi cách dựng hình theo đề bài trên thành như sau+Lấy điểm D trên trung trực BQ+Trên BC lấy điểm P sao cho tam giác BDP cân tại D+BD cắt trung trực BC tại Abài toán trở thành dựng điểm D sao cho AQ vuông góc PQ *Chứng minh, ta sẽ ch minh Q đối xứng với P qua DE, với P thuộc BC và PD//AC và PE//AB theo cách dựng trêncó DQ =DB =DP (1)kẻ đường thẳng qua D vuông góc PQ cắt AC tại E', cắt AP tại G (2)(1, 2)⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ E'D là trung trực PQ (3)từ (3) vàAQP^=90∘⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">ˆAQP=90∘⇒AQP^=90∘⇒ G là trung điểm AP⇒△GDP=△GE′A" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒△GDP=△GE′A⇒△GDP=△GE′A (g, c, g)⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ ADPE' là hình bình hành ⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ PE' //AB⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ E' trùng E (4)từ (3, 4)⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ đpcm *Giảicó B, Q⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ pt trung trực BQ là y =0⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ tọa độ D=(d,0)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">D=(d,0)D=(d,0)có A thuộc trung trực BC ⇒A=(a,2a)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒A=(a,2a)⇒A=(a,2a)BD→=(d+2,−1)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−−→BD=(d+2,−1)BD→=(d+2,−1)BA→=(a+2,2a−1)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−−→BA=(a+2,2a−1)BA→=(a+2,2a−1)có BD→" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−−→BDBD→ và BA→" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−−→BABA→ cộng tuyến⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ (d +2)(2a -1) =-(a +2)⇔" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇔⇔ (2d +5) .a =d (5)nếu 2d +5 =0⇒0=−52" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒0=−52⇒0=−52 (vô lí)⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ 2d+5≠0" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2d+5≠02d+5≠0(5)⇒a=d2d+5" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a=d2d+5⇒a=d2d+5⇒A=(d2d+5,2d2d+5)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒A=(d2d+5,2d2d+5)⇒A=(d2d+5,2d2d+5)⇒QA→=(5d+102d+5,4d+52d+5)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒−−→QA=(5d+102d+5,4d+52d+5)⇒QA→=(5d+102d+5,4d+52d+5)Hạ DF vuông góc BC tại FF thuộc BC⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒ F=(-2 +4t, 1 -2t)⇒FD→=(d+2−4t,2t−1)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒−−→FD=(d+2−4t,2t−1)⇒FD→=(d+2−4t,2t−1)FD→.BC→=0" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−−→FD.−−→BC=0FD→.BC→=0⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒4(d +2 -4t) -2(2t -1) =0⇒t=2d+510" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒t=2d+510⇒t=2d+510⇒F=(4d5,−2d5)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒F=(4d5,−2d5)⇒F=(4d5,−2d5)P đối xứng với B qua F⇒P=(8d+105,−4d+55)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒P=(8d+105,−4d+55)⇒P=(8d+105,−4d+55)QP→=(8d+205,−4d5)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−−→QP=(8d+205,−4d5)QP→=(8d+205,−4d5)có QA→.QP→=0" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−−→QA.−−→QP=0QA→.QP→=0⇒(5d+10)(8d+20)−(4d+5)4d=0" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒(5d+10)(8d+20)−(4d+5)4d=0⇒(5d+10)(8d+20)−(4d+5)4d=0⇒3d2+20d+25=0" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒3d2+20d+25=0⇒3d2+20d+25=0⇒[d=−5d=−53" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒[d=−5d=−53⇒[d=−5d=−53+d=-5⇒P=(−6,3)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒P=(−6,3)⇒P=(−6,3)⇒" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒⇒loại d =-5 vì P nằm ngoài đoạn BC+d=−53" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">−53−53⇒P=(−23,13)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒P=(−23,13)⇒P=(−23,13)P nằm giữa B, C ⇒d=−53" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒d=−53⇒d=−53 thỏa mãn⇒A=(−1,−2)" role="presentation" style="padding-top: 1px; padding-bottom: 1px; font-size: 18.06px; display: inline-block; line-height: 0; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒A=(−1,−2)Hình gửi kèm