As shown in the figure, place the right angle vertex P of the triangle plate PMN on the diagonal BD of the square ABCD, and rotate the triangle plate around point P. the two right angle sides PM and PN of the triangle plate intersect AB at e and BC at f respectively (1) Verification: PE = PF (2) The quantitative relationship among be, BF and BP is expressed by equation and explained

As shown in the figure, place the right angle vertex P of the triangle plate PMN on the diagonal BD of the square ABCD, and rotate the triangle plate around point P. the two right angle sides PM and PN of the triangle plate intersect AB at e and BC at f respectively (1) Verification: PE = PF (2) The quantitative relationship among be, BF and BP is expressed by equation and explained

(1)PE=PF.
(2)BE+BF=√2BP.
It is proved that PG is perpendicular to g, pH is perpendicular to H
If BH is perpendicular to BG, then phbg is rectangular;
And ∠ PBG = 45 ° so BG = PG, BP = √ 2BG, quadrilateral phbg is square, PG = pH = HB = BG;
If pf = PE, then: RT ⊿ Phe ≌ RT Δ PGF (HL), eh = FG
Therefore: be + BF = (bh-eh) + (BG + FG) = (bg-fg) + (BG + FG) = 2BG = √ 2 * (√ 2BG) = √ 2bp

It is known that: as shown in the figure, AC and BD are diagonals in square ABCD. Rotate ∠ BAC around vertex a anticlockwise (0 ＜ α ＜ 45). After rotation, the two sides of the angle intersect BD at points P and Q, BC, CD at points E and F, connecting EF and Eq. (1) does the size of ∠ AEQ change during the rotation of ∠ BAC? If it remains unchanged, write its degree; if it changes, write its range of change (write the result directly on the answer sheet, without proof); (2) explore the quantitative relationship between the area of △ Apq and △ AEF, write the conclusion and prove it

(1) Let the diagonals intersect at point O. from the meaning of the title, we can see that ∠ BAE = α°, ∠ oaq = α °, so ∠ BAE = ∠ oaq, because ∠ Abe = ∠ AOQ = 90 °, so △ Abe ∽ AOQ ∽ AB: Ao = AE: AQ, so AB / AE = AO / AQ, and because ∠ Bao = ∠ EAQ = 45 °, so △ Bao ∽ ea

Known: as shown in the figure, in square ABCD, AC and BD are diagonals, and ∠ BAC is rotated counterclockwise around vertex a
It is known that: as shown in the figure, AC and BD are diagonals in square ABCD, and rotate ∠ BAC counterclockwise around vertex a (0

(1) The two angles of the same side pair of the chord EQ are both 45 degrees, so the four points of abeq are in the same circle. The circumferential angles of the chord AQ pairs are equal. That is: ∠ AEQ = ∠ ABQ = 45 degrees unchanged, the area ratio of 45 degrees (2) is 1:2 △ Apq is similar to △ AEF, and a vertical line is made across a direction EF. The perpendicular foot is m, △ ame and △ Abe congruent or △ AMF and △ ADF congruent Ao: am = Ao: ab = 1