As shown in the figure, △ ABC is an equilateral triangle, P is a point on the bisector BD of ∠ ABC, PE ⊥ AB is at point E, the vertical bisector of line BP intersects BC at point F, and the perpendicular foot is point Q. if BF = 2, the length of PE is () A. 2B. 23C. 3D. 3

As shown in the figure, △ ABC is an equilateral triangle, P is a point on the bisector BD of ∠ ABC, PE ⊥ AB is at point E, the vertical bisector of line BP intersects BC at point F, and the perpendicular foot is point Q. if BF = 2, the length of PE is () A. 2B. 23C. 3D. 3

∵△ ABC is an equilateral triangle, P is the bisector of ∠ ABC, ∵ EBP = ∠ QBF = 30 °, ∵ BF = 2, QF is the vertical bisector of line BP, ∵ fqB = 90 °, ∵ BQ = BF · cos30 ° = 2 × 32 = 3, ∵ BP = 2BQ = 23, in RT △ BEP, ∵ EBP = 30 °, ∵ PE = 12bp = 3

As shown in the figure, in the triangle ABC, ab = AC, P is a point on BC, PE is perpendicular to AB and E, PE is perpendicular to AC and F, BD is perpendicular to AC and D
1. Judge the quantitative relationship among PE, PF and BD, and prove the relationship
2. If P is a little bit on the BC extension line, what do you find

Use area to solve the problem
1,PE+PF=BD
Using s △ ABP + s △ ACP = s △ ABC
2,S△ABP=S△ACP+S△ABC
Finally, we can get PE = pf + BD

In the equilateral triangle ABC, point E is on AB, point D is on the extension line of CB, and ED = EC. Try to judge the size relationship between AE and DB? Explain the reason
When passing through point E, EF is parallel to BC and AC is parallel to point F

AE＝BD
It is proved that when passing through point E, EF ‖ BC intersects AC with F
∵ equilateral △ ABC
∴∠A＝∠ABC＝60
∴∠ABD＝180-∠ABC＝120
∵EF∥BC
∴∠AEF＝∠ABC＝60,∠FEC＝∠BCE
Ψ equilateral △ AEF
∴EF＝AE,∠AFE＝60
∴∠EFC＝180-∠AFE＝120
∴∠ABD＝∠EFC
∵ED＝EC
∴∠D＝∠BCE
∴∠D＝∠FEC
∴△BDE≌△EFC （AAS）
∴EF＝BD
∴AE＝BD

In the right triangle, obtuse triangle and acute triangle, there are two high triangles outside the triangle______ A triangle

There are two obtuse triangles that are high on the outside of the triangle