Here are a couple of problems for you to practise. These problems are from older sets in Brilliant.org.
1)A circle $\Gamma$ cuts the sides of a equilateral triangle $ABC$ at $6$ distinct points. Specifically, $\Gamma$ intersects $AB$ at points $D$ and $E$ such that $A$,$D$,$E$,$B$ lie in order. $\Gamma$ intersects $BC$ at points $F$ and $G$ such that $B$,$F$,$G$,$C$ lie in order. $\Gamma$ intersects $CA$ at points $H$ and $I$ such that $C$,$H$,$I$,$A$ lie in order. If $|AD|=3$, $|DE|=39$, $|EB|=6$ and $|FG|=21$, what is the value of $|HI|^2$?
Hint: Try using the intersecting secants theorem. See the original problem here
2)Circles $\Gamma_1$ and $\Gamma_2$ have centers $X$ and $Y$ respectively. They intersect at points $A$ and $B$, such that angle $XAY$ is obtuse. The line $AX$ intersects $\Gamma_2$ again at $P$, and the line $AY$ intersects $\Gamma_1$ again at $Q$. Lines $PQ$ and $XY$ intersect at $G$, such that $Q$ lies on line segment $GP$. If $GQ=255$, $GP=266$ and $GX=190$, what is the length of $XY$?
Hint: Power-of-a-point or noticing cyclic quads could help. The original problem is here
3)$ABC$ is a triangle with $AC=139$ and $BC=178$. Points $D$ and $E$ are the midpoints of $BC$ and $AC$ respectively. Given that $AD$ and $BE$ are perpendicular to each other, what is the length of $AB$?
Hint: The centroid of a triangle trisects its medians. The original problem
4)$ABCD$ is a trapezoid(US) or Trapezium(UK) with parallel sides $AB$ and $CD$. $\Gamma$ is an inscribed circle of $ABCD$, and tangential to sides $AB$,$BC$,$CD$ and $AD$ at the points $E$,$F$,$G$ and $H$ respectively. If $AE=2$,$BE=3$, and the radius of $\Gamma$ is $12$, what is the length of $CD$?
Hint: There are multiple ways to solve this problem. Are you familiar with Pitot's Theorem? See the original problem here
Have fun solving these problems!