Why is Vbe a constant 0.7 for a transistor in the active region?
Inverting the collector current equation:

$$i_C = I_Se^{\frac{v_{BE}}{V_T}}$$

yields:

$$v_{BE} = V_T\ln{\frac{i_C}{I_S}}$$

For example, let

$$V_T = 25mV$$

$$I_S = 1 fA$$

$$I_C = 1mA$$

With these values, find that

$$V_{BE} = 0.691V$$

Now, double the collector current and find that

$$V_{BE} = 0.708V$$

Increasing the collector current 100% only increased the base-emitter voltage 2.45%

So, while it is not true that the base-emitter voltage is constant, it is not a bad approximation to consider it constant over a relatively wide range of collector current.