I got the picture from my prof's notes, and I don't quite understand why there can be two different currents at the given node. If there are only two currents shouldn't the currents' values be the same
On the top node you can see that there is one incoming current that is equal to the sum of the two outgoing currents there. At the bottom node, you can see that the two incoming currents are equal to the outgoing current. And at the middle node (base of BJT), there is one incoming current and two outgoing currents and the sum of the two outgoing equals the incoming. Finally, you can see that the node at the BJT emitter receives the collector current and the base current as incoming currents and that the load current equals the sum of those. It all works out just fine.
The current going into the node from the left (the power supply) is 1.0209A, determined by KCL.
It leaves (splits into two) as 39mA, of which about 10mA flows into the load through the base-emitter junction and 0.99A which flows into the load through the collector-emitter junction.
The signed total of all three currents at that node is 0.0 as it must be.