The quantitative relationship between (1) line segment AP and AQ: AP+AQ=3 (there is also an intersection point Q between the circle with the diameter of EC and AD, which can be proved by Pythagorean theorem).

(2) the range of be: 7∕8≤BE≤2.

Let PD=x, then AP=3-x, and the utilization similarity is AE: 3-x = x: 2.

AE= 1/2(x-3/2)？ +9/8

The maximum value of AE is 9/8,

The minimum value of BE is 7/8.