As shown in the figure, △ abd and △ ace are both RT △ where ∠ abd = ∠ ace = 90 ° C is on AB, connecting De, and M is the midpoint of de. verification: MC = MB

It is proved that the intersection of CM and dB at g, ∵ abd and △ ace are RT △ and ∵ CE ∥ BD, that is, CE ∥ DG, ∵ CEM = ∵ GDM, ∵ MCE = ∵ MGD and ∵ m is the midpoint of De, that is, DM = em, ≌ ECM ≌ DMG, ≌ cm = mg, ∵ G is on the extension line of DB, ∵ CBG is RT △ CBG, ∵ in RT △ CBG, BM = 12cg = cm

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1. As shown in the figure, △ ABC is an isosceles triangle, ∠ ACB = 90 °, extend AB to F, so that ∠ ECF = 135 °. Verification: AE: EC = Ba: CF
2. The △ ACB is an isosceles triangle, ∠ ACB = 90 ° extend Ba to e, extend AB to F, ∠ ECF = 135 ° Q: is there an equivalent relationship between AB, AE and BF
3. In △ ABC, ab = AC, CG ⊥ Ba intersects the extension line of BA at point g. an isosceles right triangle ruler is placed as shown in Figure 1. The right angle vertex of the triangle ruler is F. one right angle side is in a straight line with AC side, and the other right angle side just passes through point B (1) In Figure 1, please guess and write down the quantitative relationship between BF and CG by observing and measuring the length of BF and CG, and then prove your conjecture; (2) when the triangle ruler moves to the position shown in Figure 2 along the direction of AC, one right angle side is still on the same line with AC side, the other right angle side intersects BC side at point D, and makes de ⊥ BA at point e through point D. at this time, please observe and measure De, D The length of F and CG, conjecture and write the quantitative relationship between de + DF and CG, and then prove your conjecture; (3) whether the conjecture in (2) is still true when the triangle ruler continues to move to the position shown in Fig. 3 along the direction of AC on the basis of (2) (the point F is on the line AC, and the point F does not coincide with point C) (do not explain the reason)
4. Δ ABC is isosceles right triangle, ∠ BAC = 90 °, D is a point on AC, extend Ba to e, make AE = ad, prove BD perpendicular to CE
5. If P and Q are the triangles of the hypotenuse BC, then Tan ∠ PAQ =?
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