Give the quadrilateral ABCD. \(A_1,B_1,C_1,D_1\) is the center of the circle circumscribed to the triangle BCD, CDA, DAB, ABC. \(A_2,B_2,C_2,D_2\) is the center of the circle circumscribed to the triangle \(B_1C_1D_1,C_1D_1A_1,D_1A_1B_1,A_1B_1C_1\). Prove that : \(\frac{S_{A_2B_2C_2D_2}}{S_{ABCD}}=\frac{\left(cotA+cotC\right)^2\left(cotB+cotD\right)^2}{16}\)
Given quadrilateral ABCD. M, N are the midpoints of BC, AD. Set: AC=x, BD=y, MN=z, and \(p=\frac{x+y+2z}{2}\). Prove that: \(S_{ABCD}=\sqrt{p\left(p-x\right)\left(p-y\right)\left(p-2z\right)}\)
Give the triangle ABC have inscribed circles (I). (I) contact BC, CA, AB at D, E, F. Let M and N be the midpoints of BC and IA. DF \(\cap\) IE = {P}, DE \(\cap\) IF = {Q}. Prove that: MN \(\bot\) PQ
Chứng minh
1.\(\frac{h_a}{h_b}=\frac{sinA}{sinB}\)
2.\(cotA+cotB+cotC\ge\sqrt{3}\)
3.\(\left(b^2-c^2\right)cosA=a\left(c.cosC-b.cosB\right)\)
4.\(a^2=b^2+c^2-4S.cotA\)
5.\(a^2+b^2\ge\frac{4S}{sinC}\)
Chứng minh
\(\frac{\left(sina+cosa\right)^2-1}{cota-sina.cosa}=2tan^2a\)
cho a,b,c >0; abc=1.chứng minh
\(\frac{1}{a^3\left(b+c\right)}+\frac{1}{b^3\left(c+a\right)}+\frac{1}{c^3\left(a+b\right)}\ge\frac{3}{2}\)
Chứng minh
1.\(tanA=\frac{abc}{R\left(b^2+c^2-a^2\right)}\)
2.\(h_a=\frac{a.sinB.sinC}{sin\left(B+C\right)}\)
3.\(a\left(cosB+cosC\right)+b\left(cosC+cosA\right)+c\left(cosA+cosB\right)=2p\)
4.\(\left(b+c\right)cosA+\left(a+c\right)cosB+\left(b+a\right)cosC=a+b+c\)
cho 3 số thực dương a,b,c
CMR: \(\frac{a^4}{\left(b+c\right)^2}+\frac{b^4}{\left(a+c\right)^2}+\frac{c^4}{\left(a+b\right)^2}\ge\frac{1}{4}\left(a^2+b^2+c^2\right)\)
Cho a,b,c>0. Chứng minh rằng:
\(\frac{b^3}{a^2\left(a^3+2b^3\right)}+\frac{c^3}{b^2\left(b^3+2c^3\right)}+\frac{a^3}{c^2\left(c^3+2a^3\right)}\ge\frac{1}{3}\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\).
bài 1: cho hàm số y=\(\frac{2x-1}{x+1}\). Tìm tọa độ điểm thuộc đồ thị có tung độ bằng -1
bài 2: cho hình chữ nhật ABCD, có độ dài cạnh AB=a; BC=2a. Khi đó \(\left|\overrightarrow{DC}+2\overrightarrow{BC}\right|\) bằng ?
bài 3: tìm nghiệm S của bpt :\(\sqrt{-x^2+2x+24}\le2\left(x+1\right)\)
bài 4 tính P= cos\(\left(\frac{\pi}{2}-\alpha\right)+2sin\left(2018\pi+\alpha\right)\). Biết \(sin\alpha=\frac{-1}{2}\) và \(\frac{-\pi}{2}< \alpha< 0\)