A geometry problem by A Former Brilliant Member

$\begin{aligned} \frac{\tan x}2 + \left (\frac{\tan x}2 \right)^2+ \left (\frac{\tan x}2 \right)^3 + \left (\frac{\tan x}2 \right)^4 + \cdots & = 100 \\ \implies \frac {\frac {\tan x}2}{1-\frac {\tan x}2} & = 100 & \small \color{#3D99F6} \text{For } \left| \frac {\tan x}2 \right| < 1 \\ \frac {\tan x}2 & = 100 - 100 \frac {\tan x}2 \\ \tan x & = 200 - 100 \tan x \\ \tan x & = \frac {200}{101} & \small \color{#3D99F6} \text{Note that } \left| \frac {\tan x}2 \right| < 1 \\ \implies x & = \tan^{-1} \frac {200}{101} \\ & \approx 1.1032 \text{ rad} \\ & \approx \boxed{63.21}^\circ \end{aligned}$