As shown in the figure BD is the diameter of circle O, ab = AC, ad intersects BC at point E, AE = 2, ed = 4 The first question is to find the length of AB, the second question is to extend DB to f so that BF = Bo, connect FA, and prove that FA is tangent to circle o

As shown in the figure BD is the diameter of circle O, ab = AC, ad intersects BC at point E, AE = 2, ed = 4 The first question is to find the length of AB, the second question is to extend DB to f so that BF = Bo, connect FA, and prove that FA is tangent to circle o

1) It can be concluded that ACE is similar to BDE, and AE / ed = 1 / 2, so AC / BD = 1 / 2, so AB / BD = 1 / 2, and because BD is the diameter of circle O, the angle bad is a right angle, so the angle abd is 60 ° so AB = ad * cot angle bad = 2 * 3.2) because BF = Bo = R (radius), BF = Bo = AB, that is

As shown in the figure, a, B, C, D are the four points on ⊙ o, ab = AC, ad intersects BC at point E, AE = 2, ed = 4. Find the length of ab

⊙ in ⊙ o, ab = AC,
The arc AB = arc AC
∴∠ABC=∠D．
And ∠ BAE = ∠ DAB,
∴△ABE∽△ADB．
∴AB
AE＝AD
AB, that is, AB2 = AE · ad = 2 × 6 = 12
∴AB=2
3．

As shown in the figure, in ⊙ o, the chord AB and DC intersect at e, and AE = EC, proving that ad = BC

It is proved that in △ AED and △ CEB, the,
∠A＝∠C
AE＝EC
∠ AED = ∠ CEB, (3 points)
≌△ CEB (ASA). (4 points)
Ц AD=BC. (5 points)

During the tree planting Festival, 834 trees were planted in the two schools. Among them, the number of trees planted in Haishi middle school was 2 times less than that in Lidong middle school. How many trees were planted in each school?

Let's plant x trees in Lidong middle school,
X + (2x-3) = 834,
The solution is: x = 279,
Then 2x-3 = 2 × 279-3 = 555,
A: there are 279 trees planted in Lidong middle school and 555 trees in Haishi middle school

During the tree planting Festival, 834 trees were planted in the two schools. Among them, the number of trees planted in Haishi middle school was 2 times less than that in Lidong middle school. How many trees were planted in each school?

Let's plant x trees in Lidong middle school,
X + (2x-3) = 834,
The solution is: x = 279,
Then 2x-3 = 2 × 279-3 = 555,
A: there are 279 trees planted in Lidong middle school and 555 trees in Haishi middle school

During the tree planting Festival, 834 trees were planted in the two schools. Among them, the number of trees planted in Haishi middle school was 2 times less than that in Lidong middle school. How many trees were planted in each school?

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Proof: ∵ ab ∥ ed,
∴∠ABD=∠EDB，
∵ in △ ABC and △ EDC,
∠ABC＝∠EDC
BC＝CD
∠ACB＝∠DCE ，
∴△ABC≌△EDC，
∴AB=ED．

As shown in the figure, ab ⊥ BD at point D, AE at point C, and BC = DC

[should be ab ⊥ BD at point B, de ⊥ BD at point D]
prove:
∵AB⊥BD,DE⊥BD
∴∠ABC=∠EDC=90º
And ? ACB = ∠ ECD
BC=DC
∴⊿ABC≌⊿EDC（ASA）
∴AB=ED

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The results showed that: (1) AF, BG were connected,
∵ AC = ad, BC = be, F and G are the midpoint of DC and CE respectively,
∴AF⊥BD，BG⊥AE．
In the right triangle AFB,
∵ h is the midpoint of the hypotenuse ab,
∴FH=1
2AB．
Similarly, Hg = 1
2AB，
∴FH=HG．
（2）∵FH=BH，
∴∠HFB=∠FBH；
∵ AHF is the external angle of  BHF,
∴∠AHF=∠HFB+∠FBH=2∠BFH；
Similarly, ∠ AGH = ∠ GAH, ∠ BHG = ∠ AGH + ∠ GAH = 2 ∠ AGH,
∴∠ADB=∠ACD=∠CAB+∠ABC=∠BFH+∠AGH．
And ? DAC = 180 ° - ∠ ADB - ∠ ACD,
=180°-2∠ADB，
=180°-2（∠BFH+∠AGH），
=180°-2∠BFH-2∠AGH，
=180°-∠AHF-∠BHG，
According to the definition of flat angle, it can be concluded that: ∠ FHG = 180 ° - ∠ AHF - ∠ BHG,
∴∠FHG=∠DAC．

It is known that, as shown in the figure, AE, BD intersect at the points c, m, F, G, which are the midpoint of AD, BC and CE respectively, ab = AC, DC = de. it is proved that MF = mg

Proof: linking AF, DG
∵ AB = AC, f is the midpoint of BC ᙽ AF ⊥ BC
In the right angle △ AFD, ∵ m is the midpoint of ad ᙽ MF = 1 / 2ad (the center line on the hypotenuse of the right angle △ is equal to half of the hypotenuse)
Similarly, Mg = 1 / 2ad
∴MF=MG