The length of equilateral △ ABC side is 8, D is a moving point on AB side, passing through D as de ⊥ BC at point E, passing through e as EF ⊥ AC at point F. (1) if ad = 2, find the length of AF; (2) when ad takes what value, de = EF

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• 1. As shown in the figure, in triangle ABC and triangle def, angle a = angle D, AC = DF, AE = BD. please explain why BC is parallel to ef
• 2. It is known that the three sides of △ ABC and △ def satisfy AB ／ de = AC ／ DF = BC ／ EF, and ∠ a = 56 °, B = 84 °, then ∠ d =, ∠ f =
• 3. In △ ABC and △ def, if ∠ a = D, ∠ B = e, ab = DF = 1, BC ratio EF = 3 to 2, then de = AC=
• 4. As shown in the figure, ∠ BAC = 106 ° in △ ABC, de and FG are the vertical bisectors of AB and AC respectively, then ∠ EAG=______ ．
• 5. The middle angle of the triangle ABC is ACB = 90 ° and D is on the extension line of BC. Eg bisects vertically, BD intersects AB on e, BD intersects g, de intersects AC on F. find: e is on the vertical bisector of AF
• 6. It is known that in △ ABC, ∠ ACB = 90 °, D is a point on the extension line of BC, e is a point on AB, and on the vertical bisector eg of BD, de intersects AC with F Verification: point E is on the vertical bisector of AF
• 7. As shown in the figure, in the triangle ABC, the angle ACB is equal to 90 degrees. D is a point on the extension of BC. E is the intersection of the vertical bisector of BD with ab. de intersects AC with F Prove that e bisects AF vertically
• 8. As shown in the figure, in the triangle ABC, the angle ACB = 90 degrees, D is a point on the BC extension line, e is the intersection of BD vertical bisector and AB, and de intersects AC at point F Verification: point E is also on the vertical bisector of AF
• 9. As shown in the figure, it is known that in △ ABC, ∠ ACB = 90 °, D is a point on the extension line of BC, e is a point on AB, and it is on the vertical bisector eg of BD, When De is intersected with AC in F, AE = EF
• 10. In triangle ABC, the bisector of angle ABC and angle ACB intersects at point O, passes through point O for EF, is parallel to BC, intersects AB at point E, intersects AC at point F
• 11. As shown in the figure, point G is the center of gravity of △ ABC, and de passes through points g, de ‖ BC, CEF ‖ AB, s △ ABC = 18 to calculate the area of quadrilateral bdef
• 12. In the triangle ABC, AE bisects the angle BAC, angle c > angle B, f is on the extension line of AE, FD is perpendicular to D, try to deduce the relationship between angle EFD, angle B and angle C
• 13. In the triangle ABC, AE bisects the angle BAC (angle c is greater than angle b), f is the point above AE, FD is perpendicular to BC and D. try to deduce the quantitative relationship between angle EFD and angle B, angle C
• 14. It is known that, as shown in figure a, in △ ABC, AE bisects ∠ BAC (∠ C ＞ b), f is the upper point of AE, and FD ⊥ BC is in D. (1) try to explain: ∠ EFD = 12 (∠ C - ∠ b); (2) when F is on the extension line of AE, as shown in Figure B, other conditions remain unchanged, is the conclusion in (1) still valid? Please give reasons
• 15. In the triangle ABC, D is the midpoint of AB; AE is 2 / 3 AC; CF is 3CD; find the area ratio of EFD and ABC
• 16. Given that the triangle ABC is equal to the triangle DAF, ab = 2, AC = 4, the circumference of the triangle DEF is even, then what is the length of ef
• 17. As shown in the figure, if the vertices e, F and D of the diamond BEDF are on the edge of △ ABC, and ab = 18, AC = BC = 12, then the perimeter of the diamond is______ ．
• 18. In triangle ABC and triangle def, if AB = de and angle BAC = angle EDF, the triangle ABC is congruent to triangle def, A: AC = DF B: angle ABC = angle def C: BC = EF D: angle ACB = angle DFE
• 19. (1) It is known that in △ ABC and △ def, ab = De, BC = EF, ∠ BAC = ∠ EDF = 100?: in △ ABC ≌ △ def (2), if the condition is changed to ab = De, BC = EF, ∠ BAC = ∠ EDF = 70?, is it still true
• 20. It is known that in △ ABC and △ def, ab = De, BC = EF, ∠ BAC = ∠ EDF = 1000;