![](data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 0 72 71" width="72"></svg>)
If the lengths of the three sides of △ ABC are a, B and C respectively, and the equation a 2 + B 2 + C 2 = AB + AC + BC is satisfied, try to determine the shape of the triangle
a²＋b²＋c²＝ab＋ac＋bc
2a²＋2b²＋2c²＝2ab＋2ac＋2bc
2a²＋2b²＋2c²-2ab-2ac-2bc=0
(a-b)^2+(a-c)^2+(b-c)^2=0
Three numbers greater than or equal to 0 add up to 0, only they are equal to 0 respectively
So a = b = C
Equilateral triangle
If a, B and C are the three side lengths of △ ABC and satisfy A3 + AB2 + BC2 = B3 + A2B + ac2, then the shape of △ ABC is () A. Isosceles triangle B. Right triangle C. Isosceles triangle or right triangle D. Isosceles right triangle
∵a3+ab2+bc2=b3+a2b+ac2，
∴a3-b3-a2b+ab2-ac2+bc2=0，
（a3-a2b）+（ab2-b3）-（ac2-bc2）=0，
a2（a-b）+b2（a-b）-c2（a-b）=0，
（a-b）（a2+b2-c2）=0，
So A-B = 0 or A2 + b2-c2 = 0
So a = B or A2 + B2 = C2
Therefore, the shape of △ ABC is isosceles triangle, right triangle or isosceles right triangle
Therefore, C
It is known that in the triangle ABC, the opposite sides of angle a, angle B and angle c are respectively a, B, C, and the three sides a, B, C satisfy the equation a2-16b2-c2 + 6ab + 10bc = 0. It is proved that a + C = 2B (2) A = 3 real number C = x ⊥ 4x + 9 minimum value, pass point C as CH ⊥ AB, perpendicular foot is h, find ch
A? - 16b? - C? + 6ab + 10bc = 0A? + 6ab + 9b? - 25B? + 10bc-c? = 0 (a + 3b)? - (5b-c)? = 0 [(a + 3b) + (5b-c)], [(a + 3b) - (5b-c)] = 0 (a + 8b-c) (a-2b + C) = 0, so 8b = C-A, or 2B =
In the triangle ABC, the three sides a, B, C satisfy a? - 16b? - C? + 6ab + 10bc = 0, and prove that a + C = 2B
a²-16b²-c²+6ab+10bc=0
(a²+6ab+9b²)-(25b²-10bc+c²)=0
(a+3b)²-(5b-c)²=0
That is: (a + 3b) 2 = (5b-c) 2 because a, B and C are the three sides of a triangle, there are:
a+3b=5b-c
A + C = 2B
If the equation AC = AB holds, then the following equation may not hold: A.A = B B B.A? C = ABC c.ac + M = BC + m d.ac-b = bc-b
B is not necessarily true
It is known that a, B, C are the three side lengths of △ ABC. When A2 + C2 + 2B (b-a-c) = 0, try to judge the shape of △ ABC
∵a2+c2+2b（b-a-c）=0，
∴a2-2ab+b2+b2-2bc+c2=0
The formula is as follows: (a-b) 2 + (B-C) 2 = 0
∴a=b=c，
The △ ABC is an equilateral triangle
It is known that A.B.C is the three sides of △ ABC and satisfies the relation a 2 ＋ B = 2Ab ＋ 2ac-2b , so try to judge the shape of △ ABC Change: a 2 + B 2 should be a 2 + C 2
Equilateral triangle
The known shape of △ 2A + 2a is the edge of △ 2B + 2a, which satisfies △ 2B
2a²+2b²+2c²=2ab+2bc+2ac
Simplification: A ^ 2-2ab + B ^ 2 + A ^ 2-2ac + C ^ 2 + B ^ 2-2bc + C ^ 2 = 0
(a-b)^2+(a-c)^2+(b-a)^2=0
A=b
b=c,
A=c
It's an equilateral triangle
Given that a, B, C are the three sides of △ ABC, the shape of △ ABC can be explained by the following conditions: a 2B C 2 = 2Ab + 2BC
a²+2b²+c²=2ab+2bc
a²-2ab+b²+b²-2bc+c²=0
(a-b)²+(b-c)²=0
∴{a-b=0
b-c=0
∴a=b=c
ν Δ ABC is an equilateral triangle
It is known that a, B, C, are the three sides of △ ABC and satisfy the relation a 2 + C 2 = 2 ab + 2 BC-2 B 2, which means that △ ABC is an equilateral triangle
Because a? 2 + C? 2 = 2Ab + 2bc-2b
So a 2 + C? - 2 ab-2bc + 2B 2 = 0
So (a-b) 2 + (B-C) 2 = 0
A = B, B = C, that is, the three sides are equal
RELATED INFORMATIONS
- 1. It is known that a, B, C are the three sides of △ ABC and satisfy the relation of a 2 + C 2 = 2 ab + 2 BC-2 B 2. Try to judge the shape of a triangle
- 2. It is known that A.B.C is the length of the three sides of the triangle ABC. If the square of B + 2Ab = the square of a + 2Ac is, judge what triangle ABC belongs to Try to explain that the square of a - the square of B + the square - 2Ac of C is greater than 0
- 3. It is known that a, B and C are the side lengths of the triangle ABC, and B ^ 2 + 2Ab = C ^ 2 + 2Ac
- 4. In the application of higher one inequality, a > 0, b > 0, it is proved that 2 (radical a + radical b) ≤ a + B + 2
- 5. Prove inequality: A4 + B4 + C4 ≥ a2b2 + b2c2 + c2a2 ≥ ABC (a + B + C)
- 6. If AB + BC + Ca = 1, then the following inequality holds () A a^2+b^2+c^2>=2 B (a+b+c)^2>=3 C 1 / A + 1 / B + 1 / C > = 2 * (radical 3) D a+b+c
- 7. In the triangle ABC, if we know the square of a plus the square of B minus C = AB, then the angle c is equal to
- 8. Given that in a triangle ABC, angle a = angle B, 1 / 4 angle c = angle a, find the degree of each angle in the triangle
- 9. In the triangle ABC, if the angle B = angle c = 15 ° AB = 2cm, CD is perpendicular to the extension line of AB intersecting AB at point D, then the length of CD is?
- 10. a. B and C are positive integers and satisfy the equation ABC + AB + AC + BC + A + B + C + 1 = 2004, then what is the minimum value of a + B + C?
- 11. It is known that the three sides of the triangle ABC are a, B, C, and satisfy the relation a square + b square + C square + 50 = 6A + 8b + 10C,
- 12. If the length of the three sides of the triangle ABC is a, B, C, and satisfies the condition that the square of a + the square of B + the square of C = AB + BC + Ca, what kind of triangle is this triangle To solve problems,
- 13. If a, B, C are the three sides of △ ABC, and A2 + B2 + C2 = AB + AC + BC, then △ ABC is () A. Isosceles triangle B. Right triangle C. Equilateral triangle D. Isosceles right triangle
- 14. It is known that a, B and C are the three sides of a triangle, a = 2n 2 + 2n, B = 2n + 1, C = 2n 2 + 2n + 1 (n is a natural number greater than 1), we try to explain that △ ABC is a right triangle
- 15. Given that a, B, C are the three sides of the triangle ABC and satisfy the relation a ^ 2 + C ^ 2 = 2Ab + 2bc-2b ^ 2, try to judge the shape of the triangle ABC and explain your reasons
- 16. Given that a, B and C are the three sides of the triangle ABC, and a square C square - b square C square = a 4 power - B power, try to judge the shape of the triangle ABC
- 17. Mathematical induction to prove inequality (1 / N + 1) + (1 / N + 2) +. + (1 / 3N + 1) > 25 / 24
- 18. It is proved by mathematical induction that the nth power of 2 > 2n + 1 (n ∈ n *, n ≥ 3)
- 19. 11 + 1 + 1 - 111 = 4 move a matchstick to make the equation hold!
- 20. Please fill in the number represented by X to make the equation true (1） X÷X=X2 X=（ ） （2）X+X=2X X=( ) (3) X2=2X X=( ) (4) X2=X-X X=( ) Which prawn can help me to solve it!