Four squares aba1b1, bcb1c1, cdc1d1 and dad1a1 are made out of the four sides AB, BC, CD and Da of the quadrilateral ABCD. The centers of the four squares are m, N, P and Q respectively, connecting MP and NQ. It is proved that MP = NQ and MP is perpendicular to NQ
van Aubel's theorem
I just figured it out,
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- 1. It is known that a, B and C are three non negative numbers, and satisfy 3A + 2B + C = 5, 2A + b-3c = 1. Let m = 3A + b-7c, find the maximum and minimum of M Please give me a detailed process
- 2. The rules are: the first number is 1, the second number is 2, and the third number is 1. Generally, write a line of 1, and then insert K 2 (k = 1, 2, 3,...) between the K 1 and K + 1 1 1 Question (1) is the number 2005 1 or 2? (2) What is the sum of the first 2005 numbers?
- 3. The distance between a and B is 60 kilometers. Xiao Wang starts from a to B at 8 a.m. at the speed of 10 kilometers per hour. After a while, Xiao Li also goes from a to B at the speed of 15 kilometers per hour. Xiao Li catches up with Xiao Wang at m on the way and informs Xiao Wang to return to a immediately. Xiao Li continues to ride to B. after arriving at a and B respectively, he returns immediately When they met again, they happened to be in m place______ It's time to start
- 4. Xiao Gang started from home, crossed the top of the mountain and went to grandma's home at the foot of the mountain. He walked 19.5 kilometers. He walked 3 kilometers an hour up the mountain and 5 kilometers an hour down the mountain. He started at 5 in the morning and arrived at Grandma's home at 10:30. After playing at Grandma's home for two hours, he returned at the same speed. What time was Xiao Gang back home in the afternoon?
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- 7. In the chess game, there are 10 students from junior high school, senior high school and literature and art department. Each of them competes with the other 9 students in one game. The winner of each game gets 1 point, and the draw gets 0. The loser doesn't score. Finally, the average score of Junior high school is 4. The average score of senior high school is 3.6, and the average score of literature and art department is 9?
- 8. In general, if the rational numbers X1 and X2 represent the points A1 and A2 on the number axis, then | x2-x1 | is called the distance between points A1 and A2. Let X1 and X2 take the following groups of data respectively, and try to find the values of A1 and A2 = | x2-x1 | (1)x1=5,x2=2;(2)x1=2,x2=-5 (3)x1=6,x2=-3 (4)x1=-3,x2=-6.
- 9. A certain electronic insect falls on a certain point Ko on the number axis and starts to jump from Ko. In the first time, it jumps left by 1 unit length to K to K1. In the second time, it jumps right by K1 by 2 unit length to K2. In the third time, it jumps left by K2 by 3 unit length to K3. In the fourth time, it jumps right by K3 by 4 unit length to K4, When it falls for the 100th time, the number of k100 on the number axis is exactly 2008, so the number of Ko is ()
- 10. Known x1, X2, X3 , xn can only take one of - 2, 0, 1, and satisfy X1 + x2 + +xn=-17,x12+x22+… +Xn2 = 37, find x13 + x23 + +The value of xn3
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- 14. Fill in the following numbers: 3, (), 6,90,9,80, (), () Fill in the following numbers: 3, (), 6,90,9,80, (), ()
- 15. Natural numbers a and B are not multiples, and their product has nine divisors. The difference between their least common multiple and the greatest common divisor is 224. What is the difference between the two natural numbers?
- 16. There is a round table on which Xiaoming and Xiaofang play chess. The rules of the game are: each person gets one chess piece at a time and puts it on the table, and the two people take turns to place it. The chess pieces are not allowed to be covered or overlapped. When there is no more place to put the chess pieces on the table, the game ends. At this time, the last person to place the chess pieces wins, How to place the pieces?
- 17. There are 16 pieces in a row on the table. Party A and Party B take turns to take at least one piece at a time and at most two pieces at a time. Party A takes the first and who gets the last one is the winner. Q: who can win? What strategy should be adopted to win?
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- 19. There are string numbers: 1,2,4,7,11,16,22. How many of the first 50 numbers in this string are divided by 3 to 1?
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