This is a very interesting problem. The **easy **solution is using brute force.

**Short Answer: **1.61 (approximately); that is the **Golden Ratio**

**Long Answer: **

Suppose A= (0,a), C = (0,0), B = (b, 0).

Then AH /BH = \( \frac {a^2} {b^2} \) (why? either use geometry or brute force computation using coordinate geometry).

The fact that HP is perpendicular to AQ provides with the following equation: $$ 1 + \frac{b^2}{a^2} = \frac{a^2}{b^2} $$

(**How? **Find inradius, find the coordinates of P and Q, compute slopes of HP and AQ, set the product of the slopes equal to -1)

Assume \( \frac{a^2}{b^2} = t \) and solve for **t. **

I will upload the computations if you are unable to do it.