find the space left in a triangle with an almost inscribed circle

Question

is there a way to find the length of the gap that is left if a triangle is almost inscribed by a circle then the top bit of the circle is cut off? (sorry I'm bad at explaining stuff) Here's what I mean: is there a way to find p?

Answer 1

HINT

$e$ is gap angle balance between side and tangent to circle of length T from A. The gap

$$ g = T \tan e= \sqrt{p^2-\frac{d^2}{4}}\tan (\pi-x-y- 2 \sin^{-1} \frac{d}{2p})$$

Answer 2

In the attached diagram below, AB and AC are tangent to the circle at B and C respectively. Therefore, triangle ABC is isosceles, and since AD is perpendicular to BC, it must bisect the angle BAC, and its extension must pass through the centre of the circle at O. Now assuming that $\ p\ $ is the length of AD (this was not completely clear in the diagram cited in the question), then we have \begin{align} p&=|\text{AD}|\\ &=|\text{AC}|\cos\frac{z}{2}\\ &=|\text{OC}|\frac{\cos\frac{z}{2}}{\tan\frac{z}{2}}\\ &=\frac{d\cos\frac{z}{2}}{2\tan\frac{z}{2}}\ , \end{align} where $\ d=2|\text{OC}|\ $ is the diameter of the circle.