![](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>)
As shown in the figure, the tangent line AC passing through a point a on ⊙ O and the extension line of diameter BD of ⊙ o intersect at point C, and AE ⊥ BC passing through a intersects at point E. (1) prove: ∠ CAE = 2 ∠ B; (2) know: AC = 8 and CD = 4, find the radius of ⊙ O and the length of AE
(1) It is proved that: connecting OA, ∵ CA to ⊙ o at point a, ∵ OAC = 90 degree, that is: ∵ CAE + ∵ 1 = 90 degree, AE ⊥ BC, ∵ 2 + ∵ 1 = 90 degree, ∵ CAE = ∵ 2. OA = ob, ∵ 3 = ∵ B, ∵ 2 = 2 ∵ B, ∵ CAE = 2 ∵ (2) ∵ AC is the tangent line of ⊙ o, ∵ Ca2 = CD · CB. ∵ CB = ca2cd
RELATED INFORMATIONS
- 1. As shown in the figure, BD is the diameter of circle O, e is a point on circle O, the straight line AE intersects the extension line of BD at point a, BC ⊥ AE at point C, and ∠ CBE = ∠ DBE. (1) try to explain that AC is the tangent line of circle O. (2) if the radius of circle O is 2, AE = 4, root 2, find the length of de
- 2. As shown in the figure, points a, B, D and E are on ⊙ o, and the extension lines of strings AE and BD intersect at point C. If AB is the diameter of ⊙ o, D is the midpoint of BC. Try to judge the size relationship between AB and AC, and give the proof
- 3. The point ABCDE is on the circle O, and the extension lines of the strings AE and BD are compared with the point C. AB is the diameter of the circle and D is the midpoint of BC Judge the size relationship between AB and AC What conditions should triangle ABC satisfy? Point e must be the midpoint of AC
- 4. As shown in the figure, BC is the diameter of circle O, a is a point on circle O, cross point C to make the tangent line of circle O, intersect the extension line of BA at point D, and take the midpoint e of CD, the extension line of AE and the tangent line of BC (1) prove that AP is the tangent of circle O. (2) if OC = CP, ab = 3 √ 3, find the length of CD
- 5. As shown in Figure 7, in rectangular ABCD, AE is vertical BD, e is perpendicular foot, and DAE = 3-angle BAE 1. Find the degree of angle abo 2. Find the degree of EAC
- 6. As shown in the figure, in the rectangular ABCD, AE ⊥ BD, the perpendicular foot is e, ∠ DAE = 3 ∠ BAE, can you find out the degree of ∠ DAE and ∠ BAE? Have proof!
- 7. As shown in the figure, in rectangular ABCD, AE is perpendicular to BD, and/_ DAE:/_ Bae = 3:1/_ BAE、/_ Degree of EAO
- 8. As shown in the figure, in the rectangular ABCD, AE ⊥ BD, the perpendicular foot is e, ∠ DAE = 2 ∠ BAE, the degree of ball ∠ BAE and ∠ BDC
- 9. As shown in the figure, in the rectangular ABCD, AE ⊥ BD, e is the perpendicular foot, ∠ DAE: ∠ BAE = 1:2, then ∠ CAE=______ Degree
- 10. In the triangle ABC, angle a = 96 degrees, extend BC to D, and the bisector of angle ABC and angle ACD intersects at point A1, The bisector of angle a1bc and angle a1cd intersects at point A2, and so on, the bisector of angle ANBC and angle ancd intersects at point an + 1 Problem 1, find angle A1, angle A2, angle A3 2. Conjecture the formula of degree of angle an (n is a positive integer) 3. Use your conjectured formula to find the degree of angle an
- 11. As shown in the figure, points a, B, D and E are on ⊙ o, and the extension lines of strings AE and BD intersect at point C. If AB is the diameter of ⊙ o, D is the midpoint of BC. Try to judge the size relationship between AB and AC, and give the proof Under the above conditions, what other conditions do △ ABC need to satisfy, and point e must be the midpoint of AC?
- 12. In △ ABC, be is the middle line, ad ⊥ BC is in D, ∠ CBE = 30 °, please prove: ad = be The figure can be drawn according to the title. There are three sides of the triangle, one middle line and one high line. There are five sides in total
- 13. It is known that be is the middle line ad ⊥ BC in △ ABC, and it is shown that ad = be
- 14. In triangle ABC, ad is high, be is middle line, angle CBE = 30 degrees, prove: ad = be
- 15. In △ ABC, ∠ BAC = 90 °, BD bisection ∠ ABC, AE ⊥ BC in E. verification: AF = ad
- 16. In △ ABC, ∠ BAC = 90 °, BD bisection ∠ ABC, AE ⊥ BC in E. verification: AF = ad
- 17. In △ ABC, a-c = 90 ° AE is the bisector of BAC, and the degree of AEC is calculated
- 18. As shown in the figure, in △ ABC, ad is the height of BC, AE is divided equally into ∠ BAC, ∠ B = 75, ∠ C = 45, and ∠ Daewoo and ∠ AEC are calculated
- 19. It is known that ad is the angular bisector of △ ABC, de ∥ AC intersects AB at point E, DF ∥ AB intersects AC at point F
- 20. It is known that ad is the angular bisector of △ ABC, de ∥ AC intersects AB at point E, DF ∥ AB intersects AC at point F