请用视频解答图片上的几何题---【例9】
如图 1, 在△ABC 中, ∠B=90°, 分别作其内角 ∠ACB 与外角 ∠DAC 的平分线, 且两条角平分线所在的直线交于点 E.
(1) 猜想 ∠E 的度数, 并说明理由
(2) 分别作 ∠EAB 与 ∠ECB 的平分线, 且两条平分线交于点 F.
①依题意在图 1 中补全图形;
②直接写出 ∠AFC 的度数=___;
(3) 在 (2) 的条件下, 射线 FM 在 ∠AFC 的内部且 ∠AFM = $\frac{1}{a}$ ∠AFC(a>1), 设 EC 与 AB 的交点为 H, 射线 HN 在 ∠AHC 的内部且 ∠AHN = $\frac{1}{a}$ ∠AHC, 射线 HN 与 FM 交于点 P, 若 ∠FAH, ∠FPH 和 ∠FCH 满足的数量关系为 ∠FCH = m ∠FAH + n ∠FPH, 请直接写出 m 的值为____, n 的值为____ (用 a 表示).
Chart/Diagram Description:
Figure 1:
- Type: Geometric figure.
- Main Elements:
- Points: A, B, C, D, E.
- Lines: A right-angled triangle ABC at B. Line segment BC is horizontal, AB is vertical. Line AC connects A and C. Line extending AB upwards to D. Line CE is drawn from C. Line AE is drawn from A.
- Shapes: Right triangle ABC.
- Angles: ∠B is marked as 90 degrees. CE is labeled as the angle bisector of ∠ACB. AE is labeled as the angle bisector of ∠DAC. E is the intersection of CE and AE.
- Labels: 图 1, A, B, C, D, E.
备用图 (Spare Diagram):
- Type: Geometric figure.
- Main Elements:
- Points: A, B, C, E.
- Lines: Two lines forming a right angle at B. One line is horizontal (BC), the other is vertical (AB). Line AE is drawn from A. Line CE is drawn from C.
- Shapes: A right angle at B.
- Angles: ∠B is marked as a right angle (90°).
- Labels: 备用图, A, B, C, E.